Finite energy chiral sum rules in QCD

The saturation of QCD chiral sum rules of the Weinberg-type is analyzed using ALEPH and OPAL experimental data on the difference between vector and axial-vector correlators (V-A). The sum rules exhibit poor saturation up to current energies below the tau-lepton mass. A remarkable improvement is achieved by introducing integral kernels that vanish at the upper limit of integration. The method is used to determine the value of the finite remainder of the (V-A) correlator, and its first derivative, at zero momentum: Π̄(0) = −4L̄10 = 0.0257± 0.0003 , and Π̄ ′(0) = 0.065± 0.007 GeV. The dimension d = 6 and d = 8 vacuum condensates in the Operator Product Expansion are also determined: < O6 >= −(0.004 ± 0.001) GeV , and < O8 >= −(0.001 ± 0.006) GeV . Work supported in part by the Volkswagen Foundation Since the pioneering work of Shifman, Vainshtein and Zakharov [1], a few thousand papers have been published on applications of the QCD sum rule method in all corners of low energy hadronic physics. Unavoidably, results from different collaborations were not always consistent [2]. The main reason for these inconsistencies was frequently the impossibility of estimating reliably the errors in the method. With the advent of precise measurements of the vector (V) and axial-vector (A) spectral functions, obtained from tau-lepton decay [3]-[4], an opportunity was opened to check the precision of the QCD sum rules in the light-quark sector of QCD. In this note we would like to present a critical and conservative appraisal of chiral sum rules of the Weinberg type [5], as they are confronted with experimental data for the spectral functions. This kind of sum rules involve the difference between the vector and the axial-vector correlators (V-A), which vanishes identically to all orders in perturbative QCD in the chiral limit. In fact, neglecting the light quark masses, the (V-A) two-point function vanishes like 1/q in the space-like region, where the scale O(300 MeV) is set by the quark and gluon condensates. In the time-like region the chiral spectral function ρV −A(q ) should also vanish for large Q ≡ −q, but judging from the ALEPH data[3], the asymptotic regime of local duality may not have been reached in τ -decay . Under less stringent assumptions one expects global duality to hold in the time like region; in particular, this should be the case for the Weinberg-type sum rules. Surprisingly, these sum rules also appear to be poorly convergent. A possible source of duality violation could be some non-perturbative contribution to the correlator (e.g. due to instantons) which falls off exponentially in the space-like region but oscillates in the time-like region. If the duality violations were due to this source, then there would be a simple recipe (introduced 30 years ago [6]) to improve convergence. In a previous publication [7] we studied some QCD chiral sum rules of the Weinberg type, and their saturation by the ALEPH data. In particular, we showed that a remarkable improvement of this saturation can be achieved by introducing a polynomial integration kernel which vanishes at the upper limit of integration. However, no detailed quantitative error analysis was performed in [7]. In this note we reexamine the saturation of several QCD chiral sum rules using the ALEPH [3], as well as the OPAL data [4], and paying particular attention to the error analysis. We obtain an updated determination of L̄10, the scale independent part of the coupling constant of the relevant operator in the O(p) counter terms in the Lagrangian of chiral perturbation theory [8]. This quantity is related to the finite remainder of the (V-A) correlator at zero momentum. We also determine the finite remainder of the first derivative of the (V-A) correlator at zero momentum, which is related to the O(p) counter terms. Finally, we introduce combinations of QCD chiral sum rules which allow for a determination of the (V-A) dimension d = 6 and d = 8 vacuum condensates. The former can be extracted with reasonable precision, while the latter is affected by much larger uncertainties.