Long-distance reconstruction of QED corrections to the hadronic vacuum polarization for the muon <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>g</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:math>

The long-distance contribution of QED corrections to the hadronic vacuum polarization is particularly challenging to compute in lattice <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>QCD</a:mi> <a:mo>+</a:mo> <a:mi>QED</a:mi> </a:mrow> </a:math> . Currently, it is one of the limiting factors toward matching the precision of the recent result by the Fermilab E989 experiment for the muon <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mrow> <c:mi>g</c:mi> <c:mo>−</c:mo> <c:mn>2</c:mn> </c:mrow> </c:math> . In this work, we present a method for obtaining high-precision results for this contribution by reconstructing exclusive finite-volume state contributions. We find relations between the pion-photon contributions of individual diagrams and demonstrate the reconstruction method with lattice <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mrow> <e:mi>QCD</e:mi> <e:mo>+</e:mo> <e:mi>QED</e:mi> </e:mrow> </e:math> data at a single lattice spacing of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msup> <g:mi>a</g:mi> <g:mrow> <g:mo>−</g:mo> <g:mn>1</g:mn> </g:mrow> </g:msup> <g:mo>≈</g:mo> <g:mn>1.73</g:mn> <g:mtext> </g:mtext> <g:mtext> </g:mtext> <g:mi>GeV</g:mi> </g:math> and <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>m</i:mi> <i:mi>π</i:mi> </i:msub> <i:mo>≈</i:mo> <i:mn>275</i:mn> <i:mtext> </i:mtext> <i:mtext> </i:mtext> <i:mi>MeV</i:mi> </i:math> .