Abstract
Using ratios of the inverse Laplace transform sum rules within stability criteria for the subtraction point μ in addition to the ones of the usual τ spectral sum rule variable and continuum threshold tc, we extract the π(1300) and K(1460) decay constants to order αs4 of perturbative QCD by including power corrections up to dimension-six condensates, tachyonic gluon mass for an estimate of large order PT terms, instanton and finite width corrections. Using these inputs with enlarged generous errors, we extract, in a model-independent and conservative ways, the sum of the scale-independent renormalization group invariant (RGI) quark masses (mˆu+mˆq):q≡d,s and the corresponding running masses (m¯u+m¯q) evaluated at 2 GeV. By giving the value of the ratio mu/md, we deduce the running quark masses m¯u,d,s and condensate 〈u¯u¯〉 and the scale-independent mass ratios: 2ms/(mu+md) and ms/md. Using the positivity of the QCD continuum contribution to the spectral function, we also deduce, from the inverse Laplace transform sum rules, for the first time to order αs4, new lower bounds on the RGI masses which are translated into the running masses at 2 GeV and into upper bounds on the running quark condensate 〈u¯u¯〉. Our results summarized in Table 3 and compared with our previous results and with recent lattice averages suggest that precise phenomenological determinations of the sum of light quark masses require improved experimental measurements of the π(1.3) and K(1.46) hadronic widths and/or decay constants which are the dominant sources of errors in the analysis.
Highlights
Introduction and a short historical overviewPseudoscalar sum rules have been introduced for the first time in [1] for giving a bound on the sum of running light quark masses defined properly for the first time in the MS -scheme by [2]
The light pseudoscalar channel is quite delicate as the perturbation theory (PT) radiative corrections ([1, 15] for the αs, [13, 16] for the α2s, [17] for the α3s and [18] for the α4s corrections) are quite large for low values of Q2 ≈ 1 GeV2 where the Goldstone pion contribution is expected to dominate the spectral function, while and controversial instanton-like contributions [19,20,21] 4 might break the operator product expansion
One can still extract an optimal information on the resonance parameters if the curves present a minimum, maximum or inflexion point versus the external Laplace sum rules (LSR) variable τ and the continuum threshold tc as demonstrated in series of papers by Bell-Bertlmann [32] using the examples of harmonic oscillator and non-relativistic charmonium sum rules
Summary
Pseudoscalar sum rules have been introduced for the first time in [1] for giving a bound on the sum of running light quark masses defined properly for the first time in the MS -scheme by [2]. Some models have been proposed in the literature for parametrizing the highenergy part of the spectral function It has been proposed in [12] to extract the π(1300) and K(1460) decay constants by combining the pesudoscalar and scalar sum rules which will be used in the Laplace sum rules for extracting the light quark masses. October 5, 2018 expansion plus those beyond it such as the tachyonic gluon mass and the instanton contributions With this result, we shall extract the light quark mass values at the same approximation of the QCD series
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