Analyzing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>t</mml:mi> <mml:mover accent="true"> <mml:mi>t</mml:mi> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> <mml:mi>Z</mml:mi> </mml:math> -couplings at the fut

The proposed Large Hadron electron Collider (LHeC), with a center-of-mass energy of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msqrt> <a:mi>s</a:mi> </a:msqrt> <a:mo>≈</a:mo> <a:mn>1.3</a:mn> <a:mtext> </a:mtext> <a:mtext> </a:mtext> <a:mi>TeV</a:mi> </a:math> , provides a clean and sensitive environment to probe the top quark’s neutral current interactions with the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>Z</c:mi> </c:math> boson via <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>p</e:mi> <e:mo stretchy="false">→</e:mo> <e:msup> <e:mi>e</e:mi> <e:mo>−</e:mo> </e:msup> <e:mi>t</e:mi> <e:mover accent="true"> <e:mi>t</e:mi> <e:mo stretchy="false">¯</e:mo> </e:mover> </e:math> . We investigate the precision with which the Standard Model (SM) <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"> <j:mi>t</j:mi> <j:mover accent="true"> <j:mi>t</j:mi> <j:mo stretchy="false">¯</j:mo> </j:mover> <j:mi>Z</j:mi> </j:math> couplings—the vector and axial-vector components ( <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:mi mathvariant="normal">Δ</n:mi> <n:msub> <n:mi>C</n:mi> <n:mrow> <n:mn>1</n:mn> <n:mi>V</n:mi> </n:mrow> </n:msub> <n:mo>,</n:mo> <n:mi mathvariant="normal">Δ</n:mi> <n:msub> <n:mi>C</n:mi> <n:mrow> <n:mn>1</n:mn> <n:mi>A</n:mi> </n:mrow> </n:msub> </n:math> ) can be measured, along with possible new physics effects parametrized by higher-dimensional operators inducing weak electric and magnetic dipolelike interactions ( <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:msub> <r:mi>C</r:mi> <r:mrow> <r:mn>2</r:mn> <r:mi>V</r:mi> </r:mrow> </r:msub> <r:mo>,</r:mo> <r:msub> <r:mi>C</r:mi> <r:mrow> <r:mn>2</r:mn> <r:mi>A</r:mi> </r:mrow> </r:msub> </r:math> ). Focusing on the semileptonic decay channel, where either the top quark decays leptonically to a positively charged lepton ( <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:msup> <t:mo>ℓ</t:mo> <t:mo>+</t:mo> </t:msup> <t:mo>=</t:mo> <t:msup> <t:mi>e</t:mi> <t:mo>+</t:mo> </t:msup> <t:mo>,</t:mo> <t:msup> <t:mi>μ</t:mi> <t:mo>+</t:mo> </t:msup> </t:math> ) and the antitop hadronically, or vice versa, we utilize the azimuthal angle difference <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:mi mathvariant="normal">Δ</v:mi> <v:msub> <v:mi>ϕ</v:mi> <v:mrow> <v:msup> <v:mi>e</v:mi> <v:mo>−</v:mo> </v:msup> <v:msup> <v:mo>ℓ</v:mo> <v:mo>±</v:mo> </v:msup> </v:mrow> </v:msub> </v:math> between the scattered electron and the charged lepton <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:msup> <y:mo>ℓ</y:mo> <y:mo>±</y:mo> </y:msup> </y:math> as the key observable to derive projected constraints. Using a one-parameter multibin <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:msup> <ab:mi>χ</ab:mi> <ab:mn>2</ab:mn> </ab:msup> </ab:math> analysis of the differential <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mi mathvariant="normal">Δ</cb:mi> <cb:msub> <cb:mi>ϕ</cb:mi> <cb:mrow> <cb:msup> <cb:mi>e</cb:mi> <cb:mo>−</cb:mo> </cb:msup> <cb:msup> <cb:mo>ℓ</cb:mo> <cb:mo>±</cb:mo> </cb:msup> </cb:mrow> </cb:msub> </cb:math> distribution, the constraints on <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"> <fb:mi mathvariant="normal">Δ</fb:mi> <fb:msub> <fb:mi>C</fb:mi> <fb:mrow> <fb:mn>1</fb:mn> <fb:mi>A</fb:mi> </fb:mrow> </fb:msub> </fb:math> improve from <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" display="inline"> <ib:mi mathvariant="script">O</ib:mi> <ib:mo stretchy="false">(</ib:mo> <ib:msup> <ib:mn>10</ib:mn> <ib:mrow> <ib:mo>−</ib:mo> <ib:mn>1</ib:mn> </ib:mrow> </ib:msup> <ib:mo stretchy="false">)</ib:mo> </ib:math> at <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML" display="inline"> <nb:mn>50</nb:mn> <nb:mtext> </nb:mtext> <nb:mtext> </nb:mtext> <nb:msup> <nb:mi>fb</nb:mi> <nb:mrow> <nb:mo>−</nb:mo> <nb:mn>1</nb:mn> </nb:mrow> </nb:msup> </nb:math> to <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"> <pb:mi mathvariant="script">O</pb:mi> <pb:mo stretchy="false">(</pb:mo> <pb:msup> <pb:mn>10</pb:mn> <pb:mrow> <pb:mo>−</pb:mo> <pb:mn>2</pb:mn> </pb:mrow> </pb:msup> <pb:mo stretchy="false">)</pb:mo> </pb:math> at <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"> <ub:mn>1000</ub:mn> <ub:mtext> </ub:mtext> <ub:mtext> </ub:mtext> <ub:msup> <ub:mi>fb</ub:mi> <ub:mrow> <ub:mo>−</ub:mo> <ub:mn>1</ub:mn> </ub:mrow> </ub:msup> </ub:math> , while the improvement for <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML" display="inline"> <wb:mi mathvariant="normal">Δ</wb:mi> <wb:msub> <wb:mi>C</wb:mi> <wb:mrow> <wb:mn>1</wb:mn> <wb:mi>V</wb:mi> </wb:mrow> </wb:msub> </wb:math> remains within the same order of magnitude,