<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>K</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>π</mml:mi> <mml:msup> <mml:mi>γ</mml:mi> <mml:mo>*</mml:mo> </mml:msup> <mml:msup> <mml:mi>γ</mml:mi> <mml:mo>*<

The transition amplitude of a kaon to a pion and two off-shell photons is studied. First, it is computed at leading order (one-loop level) of the chiral perturbation theory expansion. Explicit analytical results for the leading-order amplitude are presented, constituting the first complete calculation for the doubly off-shell case. Subsequently, it is reevaluated by employing a refined diagrammatic notation and a generic ansatz incorporating effects beyond leading order. The dependence on the underlying <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>K</a:mi> <a:mi>π</a:mi> <a:mi>P</a:mi> <a:mi>P</a:mi> </a:math> vertex parameters is analyzed. This offers valuable insights into amplitude properties and allows inclusion of unitarity corrections from <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>K</c:mi> <c:mo stretchy="false">→</c:mo> <c:mn>3</c:mn> <c:mi>π</c:mi> </c:math> , yielding the complete <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>K</f:mi> <f:mo stretchy="false">→</f:mo> <f:mi>π</f:mi> <f:msup> <f:mi>γ</f:mi> <f:mo>*</f:mo> </f:msup> <f:msup> <f:mi>γ</f:mi> <f:mo>*</f:mo> </f:msup> </f:math> amplitude structure. Both the charged and neutral channels are treated in parallel. The presented results provide crucial input for phenomenological studies of related rare decays like <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>K</i:mi> <i:mo stretchy="false">→</i:mo> <i:mi>π</i:mi> <i:msup> <i:mo>ℓ</i:mo> <i:mo>+</i:mo> </i:msup> <i:msup> <i:mo>ℓ</i:mo> <i:mo>−</i:mo> </i:msup> <i:mo stretchy="false">[</i:mo> <i:mi>γ</i:mi> <i:mo stretchy="false">]</i:mo> </i:math> or <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mi>K</n:mi> <n:mo stretchy="false">→</n:mo> <n:mi>π</n:mi> <n:msubsup> <n:mo>ℓ</n:mo> <n:mn>1</n:mn> <n:mo>+</n:mo> </n:msubsup> <n:msubsup> <n:mo>ℓ</n:mo> <n:mn>1</n:mn> <n:mo>−</n:mo> </n:msubsup> <n:msubsup> <n:mo>ℓ</n:mo> <n:mn>2</n:mn> <n:mo>+</n:mo> </n:msubsup> <n:msubsup> <n:mo>ℓ</n:mo> <n:mn>2</n:mn> <n:mo>−</n:mo> </n:msubsup> </n:math> and support ongoing precision measurements at experiments like NA62 at CERN. These results may also find application in other related processes, including <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msup> <q:mi>η</q:mi> <q:mrow> <q:mo stretchy="false">(</q:mo> <q:mo>′</q:mo> <q:mo stretchy="false">)</q:mo> </q:mrow> </q:msup> </q:math> decays.