Revisit of the electromagnetic correction to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>τ</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>π</mml:mi> <mml:mi>π</mml:mi> <mml:msub> <mml:mrow> <mml

In this work we focus on the evaluation of the leading-order hadronic vacuum polarization contribution from the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>π</a:mi> <a:mi>π</a:mi> </a:math> channel to the muon anomalous magnetic moment <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>a</c:mi> <c:mi>μ</c:mi> </c:msub> </c:math> by using the experimental <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>τ</e:mi> <e:mo stretchy="false">→</e:mo> <e:mi>π</e:mi> <e:mi>π</e:mi> <e:msub> <e:mi>ν</e:mi> <e:mi>τ</e:mi> </e:msub> </e:math> data. The isospin breaking corrections play the decisive role in this approach of computing <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:msub> <h:mi>a</h:mi> <h:mi>μ</h:mi> </h:msub> </h:math> . One of such important isospin breaking sources is the long-distance electromagnetic correction factor <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:msub> <j:mi>G</j:mi> <j:mrow> <j:mi>EM</j:mi> </j:mrow> </j:msub> </j:math> of the <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mi>τ</l:mi> <l:mo stretchy="false">→</l:mo> <l:mi>π</l:mi> <l:mi>π</l:mi> <l:msub> <l:mi>ν</l:mi> <l:mi>τ</l:mi> </l:msub> </l:math> process from the real photon radiation. The latter effect can be calculated from the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>τ</o:mi> <o:mo stretchy="false">→</o:mo> <o:mi>π</o:mi> <o:mi>π</o:mi> <o:msub> <o:mi>ν</o:mi> <o:mi>τ</o:mi> </o:msub> <o:mi>γ</o:mi> </o:math> amplitude, which is revised in this work within the resonance chiral theory by simultaneously including the even-intrinsic-parity and odd-intrinsic-parity resonance operators. We update the determination of the only unknown resonance coupling through the <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:mi>ω</r:mi> <r:mo stretchy="false">→</r:mo> <r:msup> <r:mi>π</r:mi> <r:mn>0</r:mn> </r:msup> <r:msup> <r:mi>π</r:mi> <r:mn>0</r:mn> </r:msup> <r:mi>γ</r:mi> </r:math> decay by including contributions from both the vector and scalar resonances. By taking other remaining contributions from the muon <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mi>g</u:mi> <u:mo>−</u:mo> <u:mn>2</u:mn> </u:math> White Paper 2025, we further revise the complete value of <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:msub> <w:mi>a</w:mi> <w:mi>μ</w:mi> </w:msub> </w:math> , which turns out to deviate from the newest world average result after Fermilab’s measurement at the level of <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:mrow> <y:mn>2.7</y:mn> <y:mi>σ</y:mi> </y:mrow> </y:math> .