Anatomy of finite-volume effects on the hadronic vacuum polarization contribution to 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 low-energy spectrum relevant to the lattice calculation of the hadronic vacuum polarization contribution to muon anomalous magnetic moment <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msubsup><a:mi>a</a:mi><a:mi>μ</a:mi><a:mrow><a:mi>HVP</a:mi><a:mo>,</a:mo><a:mi>LO</a:mi></a:mrow></a:msubsup></a:math> is dominantly given by two-pion states satisfying Lüscher’s finite-volume quantization condition. Finite-volume effects from those states may exhibit power-law dependence on the volume, contrary to an exponential suppression as suggested by chiral effective theory. Employing the finite-volume state decomposition of Euclidean correlators, we systematically investigate the volume dependence and identify the different volume scalings depending on the volume and Euclidean-time region. Using phenomenological inputs for <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>π</c:mi><c:mi>π</c:mi></c:math> phase shift and the timelike pion form factor, we obtain an estimate for the finite-volume effects on <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msubsup><e:mi>a</e:mi><e:mi>μ</e:mi><e:mrow><e:mi>HVP</e:mi><e:mo>,</e:mo><e:mi>LO</e:mi></e:mrow></e:msubsup></e:math>, which is consistent with previous works. Numerical results are given for the “window” observables of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>a</g:mi><g:mi>μ</g:mi></g:msub></g:math>.