Abstract
The QCD up- and down-quark masses are determined from an optimized QCD Finite Energy Sum Rule (FESR) involving the correlator of axial-vector current divergences. In the QCD sector this correlator is known to five loop order in perturbative QCD (PQCD), together with non-perturbative corrections from the quark and gluon condensates. This FESR is designed to reduce considerably the systematic uncertainties arising from the hadronic spectral function. The determination is done in the framework of both fixed order and contour improved perturbation theory. Results from the latter, involving far less systematic uncertainties, are: {overline{m}}_uleft(2 mathrm{GeV}right)=left(2.6pm 0.4right) MeV, {overline{m}}_dleft(2 mathrm{GeV}right)=left(5.3pm 0.4right) MeV, and the sum {overline{m}}_{ud}equiv left({overline{m}}_u+{overline{m}}_dright)/2 , is {overline{m}}_{ud}left(2 mathrm{GeV}right)=left(3.9pm 0.3right) MeV.
Highlights
In contour improved perturbation theory (CIPT) the strong coupling is running and the renormalization group (RG) improvement is used before integration
In a variety of applications either both methods give similar results, or CIPT leads to more accurate predictions. The latter will turn out to be the case in this determination. This determination represents a substantial improvement on the previous Finite Energy Sum Rule (FESR) results for the up- and down- quark masses, in terms of (i) the analysis of different kernels, (ii) examining the issue of the convergence of the perturbative QCD expansion, (iii) a different implementation of the running QCD coupling, (iv) a more careful error analysis, and (v) the high numerical precision achieved in this calculation
The previous determination [9] performed the calculation of the quark masses in the framework of CIPT and restricted the choice of kernel to vanish at the resonance peaks, eventually preferring the kernel P5(s) = 1 − a0 s − a1 s2, with a0 = 0.897 GeV−2 and a1 = −0.1806 GeV−4
Summary
The PQCD result for ψ5 (q2) was obtained in [13], which for three flavours leads to the simplified (renormalization group improved) expression [3, 9]. As mentioned in the Introduction, the strong coupling is expressed in terms of a given scale s = s∗ where its value is known with high precision. Using the renormalization group equation for as(s) ≡ αs(s)/π one can perform a Taylor expansion at some given reference scale s = s∗, leading to [16]–[17]. By solving the renormalization group equation for m (s), the quark mass can be expressed in terms of its value at some scale s = s∗ [16, 21].
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