Finite-volume formalism for physical processes with an electroweak loop integral

This study investigates finite-volume effects in physical processes that involve the combination of long-range hadronic matrix elements with electroweak loop integrals. We adopt the approach of implementing the electroweak part as the infinite-volume version, which is denoted as the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mrow><a:mi>EW</a:mi></a:mrow><a:mi>∞</a:mi></a:msub></a:math> method in this work. A general approach is established for correcting finite-volume effects in cases where the hadronic intermediate states are dominated by either a single particle or two particles. For the single-particle case, this work derives the infinite volume reconstruction method from a new perspective. For the two-particle case, we provide the correction formulas for power-law finite-volume effects and unphysical terms with exponentially divergent time dependence. The finite-volume formalism developed in this study has broad applications, including the QED corrections in various processes and the two-photon exchange contribution in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>K</c:mi><c:mi>L</c:mi></c:msub><c:mo stretchy="false">→</c:mo><c:msup><c:mi>μ</c:mi><c:mo>+</c:mo></c:msup><c:msup><c:mi>μ</c:mi><c:mo>−</c:mo></c:msup></c:math> or <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mi>η</f:mi><f:mo stretchy="false">→</f:mo><f:msup><f:mi>μ</f:mi><f:mo>+</f:mo></f:msup><f:msup><f:mi>μ</f:mi><f:mo>−</f:mo></f:msup></f:math> decays.