Perturbative QCD below charm threshold: Theory and tensions with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>e</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msup> <mml:mi>e</mml:mi> <mml:mo>−</mml:mo>

In this paper we carefully assess the theory prediction for <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>R</a:mi> <a:mo stretchy="false">(</a:mo> <a:mi>s</a:mi> <a:mo stretchy="false">)</a:mo> </a:math> below charm threshold, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>R</e:mi> <e:mrow> <e:mi>u</e:mi> <e:mi>d</e:mi> <e:mi>s</e:mi> </e:mrow> </e:msub> </e:math> , and address tensions with the existing data, notably with the 2021 BES-III results. We analyze the uncertainty of the perturbative quantum chromodynamics description in the light of renormalons making use of the large- <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>β</g:mi> <g:mn>0</g:mn> </g:msub> </g:math> limit and the renormalon-free gluon-condensate scheme. We provide a reliable estimate of the duality violation contributions; we show they are sizable up to 2.5 GeV and improve the agreement between theory and data, but are negligible for higher energies. We then combine the available experimental data for <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>R</i:mi> <i:mrow> <i:mi>u</i:mi> <i:mi>d</i:mi> <i:mi>s</i:mi> </i:mrow> </i:msub> </i:math> and find the datasets to be mutually compatible. Finally, we compare theory and data, both locally and in their contributions to the anomalous magnetic moment of the muon. Theory is compatible with the combined data but discrepancies with the BES-III data reach more than <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mrow> <k:mn>3</k:mn> <k:mi>σ</k:mi> </k:mrow> </k:math> .