Calculating the two-photon exchange contribution to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>K</mml:mi><mml:mi>L</mml:mi></mml:msub><mml:mo stretchy="false">→</mml:mo><mml:msup><mml:mi>μ</mml:mi><mml

We present a theoretical framework within which both the real and imaginary parts of the complex, two-photon exchange amplitude contributing to <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>K</a:mi><a:mi mathvariant="normal">L</a:mi></a:msub><a:mo stretchy="false">→</a:mo><a:msup><a:mi>μ</a:mi><a:mo>+</a:mo></a:msup><a:msup><a:mi>μ</a:mi><a:mo>−</a:mo></a:msup></a:math> decay can be calculated using lattice quantum chromodynamics. The real part of this two-photon amplitude is of approximately the same size as that coming from a second-order weak strangeness-changing neutral-current process. Thus a test of the standard model prediction for this second-order weak process depends on an accurate result of this two-photon amplitude. A limiting factor of our proposed method comes from low-energy three-particle <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>π</e:mi><e:mi>π</e:mi><e:mi>γ</e:mi></e:math> states. The contribution from these states will be significantly distorted by the finite volume of our calculation—a distortion for which there is no available correction. However, a simple estimate of the contribution of these three-particle states suggests their contribution to be at most a few percent allowing their neglect in a lattice calculation with a 10% target accuracy. Published by the American Physical Society 2024