Study the property of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>W</mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:math> at future <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inli

As a strong candidate for new physics beyond the Standard Model, the exotic charged gauge boson <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mi>W</a:mi> <a:mo>′</a:mo> </a:msup> </a:math> has attracted extensive research interest. In this work, we investigate the interactions of the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msup> <c:mi>W</c:mi> <c:mo>′</c:mo> </c:msup> </c:math> boson at the electron-proton colliders. The process <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msup> <e:mi>e</e:mi> <e:mo>−</e:mo> </e:msup> <e:mi>u</e:mi> <e:mo stretchy="false">→</e:mo> <e:msub> <e:mi>ν</e:mi> <e:mi>e</e:mi> </e:msub> <e:mi>d</e:mi> </e:math> and <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:msup> <h:mi>e</h:mi> <h:mo>−</h:mo> </h:msup> <h:mi>u</h:mi> <h:mo stretchy="false">→</h:mo> <h:msup> <h:mi>e</h:mi> <h:mo>±</h:mo> </h:msup> <h:mi>j</h:mi> <h:mi>j</h:mi> <h:mi>j</h:mi> </h:math> with <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>t</k:mi> </k:math> -channel <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msup> <m:mi>W</m:mi> <m:mo>′</m:mo> </m:msup> </m:math> exchange are studied. The polarization of the initial-state electrons has a significant impact on the cross section of the studied process, while the angular distribution of the final-state leptons serves as an important observable for the interactions of the <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msup> <o:mi>W</o:mi> <o:mo>′</o:mo> </o:msup> </o:math> boson. In some specific regions of the parameter space, the detectable mass range for the <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msup> <q:mi>W</q:mi> <q:mo>′</q:mo> </q:msup> </q:math> boson can reach around 10 TeV, and the coupling strength can achieve a precision of approximately 1% relative to the interaction strength of the Standard Model. Especially, <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msup> <s:mi>e</s:mi> <s:mo>−</s:mo> </s:msup> <s:mi>u</s:mi> <s:mo stretchy="false">→</s:mo> <s:msup> <s:mi>e</s:mi> <s:mo>+</s:mo> </s:msup> <s:mi>j</s:mi> <s:mi>j</s:mi> <s:mi>j</s:mi> </s:math> process is forbidden within the Standard Model, which would constitute important evidence in the search for the left-right symmetric model.