On the use of the operator product expansion in finite-energy sum rules for light-quark correlators

Tau-based finite-energy sum-rule (FESR) analyses often assume that scales s_0\sim m_\tau^2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>∼</mml:mo> <mml:msubsup> <mml:mi>m</mml:mi> <mml:mi>τ</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> are large enough that (i) integrated duality violations (DVs) can be neglected, and (ii) contributions from non-perturbative OPE condensates of dimension D <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>D</mml:mi> </mml:math> scale as (\Lambda_{QCD}/m_\tau )^D <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:msub> <mml:mi>Λ</mml:mi> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mi>C</mml:mi> <mml:mi>D</mml:mi> </mml:mrow> </mml:msub> <mml:mi>/</mml:mi> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>τ</mml:mi> </mml:msub> <mml:msup> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>D</mml:mi> </mml:msup> </mml:mrow> </mml:math> , allowing the OPE series to be truncated at low dimension. The latter assumption is not necessarily valid since the OPE series is not convergent, while the former is open to question given experimental results for the electromagnetic, I=1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> vector ( V <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>V</mml:mi> </mml:math> ), I=1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> axial vector ( A <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>A</mml:mi> </mml:math> ) and I=1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> V+A <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> current spectral functions, which show DV oscillations with amplitudes comparable in size to the corresponding \alpha_s <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> -dependent perturbative contributions at s\sim2-3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>∼</mml:mo> <mml:mn>2</mml:mn> <mml:mo>−</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> GeV ^2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi/> <mml:mn>2</mml:mn> </mml:msup> </mml:math> . Here, we discuss recently introduced new tools for assessing the numerical relevance of omitted higher- D <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>D</mml:mi> </mml:math> OPE contributions. Applying these to the “truncated OPE” strategy used in Refs.[M. Davier et al., Eur. Phys. J. C 74, 2803 (2014); A. Pich and A. Rodríguez-Sánchez, Phys. Rev. D 94, 034027 (2016)] and earlier work by the same authors, we find that this strategy fails to yield reliable results for the strong coupling from hadronic \tau <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>τ</mml:mi> </mml:math> decays.