Strong coupling from hadronic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>τ</mml:mi></mml:math>-decay data including <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>τ</mml:mi><mm

In previous work we have combined the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mi>π</a:mi><a:mo>−</a:mo></a:msup><a:msup><a:mi>π</a:mi><a:mn>0</a:mn></a:msup></a:math>, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mn>2</c:mn><c:msup><c:mi>π</c:mi><c:mo>−</c:mo></c:msup><c:msup><c:mi>π</c:mi><c:mo>+</c:mo></c:msup><c:msup><c:mi>π</c:mi><c:mn>0</c:mn></c:msup></c:math>, and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi>π</e:mi><e:mo>−</e:mo></e:msup><e:mn>3</e:mn><e:msup><e:mi>π</e:mi><e:mn>0</e:mn></e:msup></e:math> spectral data obtained from hadronic <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>τ</g:mi></g:math> decays measured by the ALEPH and OPAL experiments, together with electroproduction data for several of the subleading hadronic modes and data for the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>K</i:mi><i:mover accent="true"><i:mi>K</i:mi><i:mo stretchy="false">¯</i:mo></i:mover></i:math> mode to construct an inclusive nonstrange vector spectral function entirely based on experimental data, with no Monte-Carlo generated input. In this paper, we include, for the first time, the Belle <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>τ</m:mi><m:mo stretchy="false">→</m:mo><m:msup><m:mi>π</m:mi><m:mo>−</m:mo></m:msup><m:msup><m:mi>π</m:mi><m:mn>0</m:mn></m:msup><m:msub><m:mi>ν</m:mi><m:mi>τ</m:mi></m:msub></m:math> high-statistics decay data to construct a new inclusive nonstrange vector spectral function that combines more of the world’s available data. As no Belle data are at present available for the two <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mn>4</p:mn><p:mi>π</p:mi></p:math> modes, this requires a revised data analysis in comparison with our previous work. From the resulting new spectral function, we obtain a new determination of the strong coupling, <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"><r:msub><r:mi>α</r:mi><r:mi>s</r:mi></r:msub></r:math>, using our previously developed strategy based on finite-energy sum rules. We find, at the <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"><t:mi>Z</t:mi></t:math> mass scale, <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"><v:msub><v:mi>α</v:mi><v:mi>s</v:mi></v:msub><v:mo stretchy="false">(</v:mo><v:msubsup><v:mi>m</v:mi><v:mi>Z</v:mi><v:mn>2</v:mn></v:msubsup><v:mo stretchy="false">)</v:mo><v:mo>=</v:mo><v:mn>0.1159</v:mn><v:mo stretchy="false">(</v:mo><v:mn>14</v:mn><v:mo stretchy="false">)</v:mo></v:math>. We discuss the smaller central value and larger error of our new result compared to our previous result, showing the shifts to be due mainly to significant changes in updated HFLAV results for the <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"><bb:msup><bb:mi>π</bb:mi><bb:mo>−</bb:mo></bb:msup><bb:mn>3</bb:mn><bb:msup><bb:mi>π</bb:mi><bb:mn>0</bb:mn></bb:msup></bb:math> decay mode.