Probing the couplings of an axionlike particle with leptons via three-lepton final state processes at future <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mro

The axionlike particle (ALP) is one of the best motivated particles beyond the Standard Model (SM). We explore the possibility of detecting the couplings of ALP with leptons via three-lepton final state processes <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:mi>p</a:mi><a:mo stretchy="false">→</a:mo><a:msup><a:mi>e</a:mi><a:mo>−</a:mo></a:msup><a:mi>j</a:mi><a:mi>a</a:mi><a:mo stretchy="false">(</a:mo><a:mi>a</a:mi><a:mo stretchy="false">→</a:mo><a:msup><a:mo>ℓ</a:mo><a:mo>+</a:mo></a:msup><a:msup><a:mo>ℓ</a:mo><a:mo>−</a:mo></a:msup><a:mo stretchy="false">)</a:mo></a:math> at the LHeC (FCC-eh). For completeness, we investigate the cases where the ALP decays not only into electron and muon pairs but also into tau pairs, and perform a detailed simulation for the decays of taus. We find that the prospective sensitivities of the LHeC (FCC-eh) with the center-of-mass energy <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msqrt><g:mi>s</g:mi></g:msqrt><g:mo>=</g:mo><g:mn>1.3</g:mn><g:mo stretchy="false">(</g:mo><g:mn>3.5</g:mn><g:mo stretchy="false">)</g:mo><g:mtext> </g:mtext><g:mtext> </g:mtext><g:mi>TeV</g:mi></g:math> and the integrated luminosity <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi mathvariant="script">L</k:mi><k:mo>=</k:mo><k:mn>1</k:mn><k:mo stretchy="false">(</k:mo><k:mn>2</k:mn><k:mo stretchy="false">)</k:mo><k:mtext> </k:mtext><k:mtext> </k:mtext><k:mrow><k:msup><k:mrow><k:mi>ab</k:mi></k:mrow><k:mrow><k:mo>−</k:mo><k:mn>1</k:mn></k:mrow></k:msup></k:mrow></k:math> to the ALP-lepton couplings are not only stronger than some existing bounds of the LHC and LEP but also complementary to the expected bounds of the CEPC and FCC-ee.