Operator product expansion with analytic QCD in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>τ</mml:mi></mml:math>decay physics

Abstract

We apply a recently constructed model of analytic QCD in the operator product expansion (OPE) analysis of the $\ensuremath{\tau}$ lepton decay data in the $V+A$ channel. The model has the running coupling ${\mathcal{A}}_{1}({Q}^{2})$ with no unphysical singularities, i.e., it is analytic. It differs from the corresponding perturbative QCD coupling $a({Q}^{2})$ at high squared momenta $|{Q}^{2}|$ by terms $\ensuremath{\propto}(1/{Q}^{2}{)}^{5}$; hence it does not contradict the OPE philosophy of ITEP School (Institute of Theoretical and Experimental Physics) and can be consistently applied with OPE up to terms of dimension $D=8$. In evaluations for the Adler function we use a Pad\'e-related renormalization-scale-independent resummation, applicable in any analytic QCD model. Applying the Borel sum rules in the ${Q}^{2}$ plane along rays of the complex Borel scale and comparing with ALEPH data of 1998, we obtain the gluon condensate value $⟨({\ensuremath{\alpha}}_{s}/\ensuremath{\pi}){G}^{2}⟩=0.0055\ifmmode\pm\else\textpm\fi{}0.0047\text{ }\text{ }{\mathrm{GeV}}^{4}$. Consideration of the $D=6$ term gives us the result $⟨{O}_{6}^{(V+A)}⟩=(\ensuremath{-}0.5\ifmmode\pm\else\textpm\fi{}1.1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}\text{ }\text{ }{\mathrm{GeV}}^{6}$, not incompatible with positive values. The real Borel transform then gives us, for the central values of the two condensates, a good agreement with the experimental results in the entire considered interval of the Borel scales ${M}^{2}$. In perturbative QCD in the minimal subtraction scheme we deduce similar results for the gluon condensate, $0.0059\ifmmode\pm\else\textpm\fi{}0.0049\text{ }\text{ }{\mathrm{GeV}}^{4}$, but the value of the $D=6$ condensate is negative, $⟨{O}_{6}^{(V+A)}⟩=(\ensuremath{-}1.8\ifmmode\pm\else\textpm\fi{}0.9)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}\text{ }\text{ }{\mathrm{GeV}}^{6}$, and the resulting real Borel transform for the central values is close to the lower bound of the experimental band.