Searching for the lepton-flavor violating <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>γ</mml:mi> <mml:mi>γ</mml:mi> <mml:mi>e</mml:mi> <mml:mi>μ</mml:mi> </mml:math> interaction at future <mml:math xmlns:mml="ht

We investigate the lepton flavor violating (LFV) process <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mi>e</a:mi> <a:mo>+</a:mo> </a:msup> <a:msup> <a:mi>e</a:mi> <a:mo>−</a:mo> </a:msup> <a:mo stretchy="false">→</a:mo> <a:msup> <a:mi>e</a:mi> <a:mo>+</a:mo> </a:msup> <a:msup> <a:mi>e</a:mi> <a:mo>−</a:mo> </a:msup> <a:mi>e</a:mi> <a:mi>μ</a:mi> </a:math> ( <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mi>e</d:mi> <d:mo>=</d:mo> <d:msup> <d:mi>e</d:mi> <d:mo>−</d:mo> </d:msup> <d:mo>,</d:mo> <d:msup> <d:mi>e</d:mi> <d:mo>+</d:mo> </d:msup> <d:mo>,</d:mo> <d:mi>μ</d:mi> <d:mo>=</d:mo> <d:msup> <d:mi>μ</d:mi> <d:mo>−</d:mo> </d:msup> <d:mo>,</d:mo> <d:msup> <d:mi>μ</d:mi> <d:mo>+</d:mo> </d:msup> </d:math> ) at future circular colliders, probing the effective <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>γ</f:mi> <f:mi>γ</f:mi> <f:mi>e</f:mi> <f:mi>μ</f:mi> </f:math> interaction through photon fusion. Within an effective field theory framework compatible with theoretical and experimental constraints, we identify regions in the parameter space of the effective couplings where the signal could be observed at the Circular Electron-Positron Collider at <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:msqrt> <h:mi>s</h:mi> </h:msqrt> <h:mo>=</h:mo> <h:mn>240</h:mn> <h:mtext> </h:mtext> <h:mtext> </h:mtext> <h:mi>GeV</h:mi> </h:math> and the Future Circular Collider (FCC-ee) at <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:msqrt> <j:mi>s</j:mi> </j:msqrt> <j:mo>=</j:mo> <j:mn>240</j:mn> </j:math> and 350 GeV. Our analysis leverages distinctive kinematic distributions—particularly the transverse momentum of the scattered electron <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:msubsup> <l:mi>p</l:mi> <l:mi>T</l:mi> <l:mrow> <l:mo stretchy="false">(</l:mo> <l:msup> <l:mi>e</l:mi> <l:mo>−</l:mo> </l:msup> <l:mi>scattered</l:mi> <l:mo stretchy="false">)</l:mo> </l:mrow> </l:msubsup> </l:math> and the central muon <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:msubsup> <p:mi>p</p:mi> <p:mi>T</p:mi> <p:mi>μ</p:mi> </p:msubsup> </p:math> —to achieve efficient signal-background separation. By employing a neural network classifier, we enhance the sensitivity beyond traditional cut-based methods, demonstrating the discovery potential of these facilities for LFV searches in the clean environment of <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:msup> <r:mi>e</r:mi> <r:mo>+</r:mo> </r:msup> <r:msup> <r:mi>e</r:mi> <r:mo>−</r:mo> </r:msup> </r:math> collisions.