**4**, 693-704

DOI 10.1007/s100529800803

## Broad sub-continuum resonances and the case

for
finite-energy sum-rules

**A.S. Deakin ^{1}^{} - V. Elias^{1}^{} - A.H.
Fariborz^{1}^{} - Ying Xue^{1}^{} -
Fang Shi^{2}^{} - T.G. Steele^{2}^{}**

^{1} Department of Applied
Mathematics, The University of Western Ontario, London, Ontario
N6A 5B7, Canada

^{2} Department of Physics and
Engineering Physics, University of Saskatchewan, Saskatoon,
Saskatchewan S7N 5C6, Canada

Received: 15 September 1997 / Revised version: 15 November 1997 / Published online: 26 February 1998

**Abstract**

There is a need to go beyond the narrow resonance approximation for
QCD sum-rule channels which are likely to exhibit sensitivity to
broad resonance structures.
We discuss how the first two Laplace sum rules are altered when one
goes beyond the narrow resonance approximation to include possible
subcontinuum resonances with nonzero widths. We show that the
corresponding first two finite energy sum rules are insensitive to
the widths of such resonances, provided their peaks are symmetric and
entirely below the continuum threshold. We also discuss the reduced
sensitivity of the first two finite energy sum rules to higher
dimensional condensates, and show these sum rules to be insensitive
to dimension > 6 condensates containing at least one pair. We extract the direct single-instanton contribution to the
*F _{1}* sum rule for the longitudinal component of the axial-vector
correlation function from the known single-instanton contribution to
the lowest Laplace sum rule for the pseudoscalar channel. Finally,
we demonstrate how inclusion of this instanton contribution to the
finite-energy sum rule leads to both a lighter quark mass and to more
phenomenologically reasonable higher-mass-resonance contributions
within the pseudoscalar channel.

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