We present the calculation of the complete NLO corrections to the off-shell top-quark pair production in the ℓ + j decay channel, denoted as \( pp\to \ell -{\overline{\nu}}_{\ell }{j}_b{j}_b jj+X \) , where ℓ− = e−, μ−. The calculation consistently preserves the finite-width effects of the top quarks and massive gauge bosons, as well as takes into account all doubly-, singly-, and non-resonant contributions along with their interference effects. All Born-level contributions, at the perturbative orders from \( \mathcal{O}\left({\alpha}_s^4{\alpha}^2\right) \) to \( \mathcal{O}\left({\alpha}_s^0{\alpha}^6\right) \) , are included and corrected by both NLO QCD and NLO EW effects. Furthermore, all possible partonic initial states are taken into account. Particular attention is paid to the infrared safety in the presence of photons and jets. This requires the use of the so-called parton-to-photon fragmentation function and the photon-to-jet conversion function, which makes the democratic photon-parton clustering and the \( \gamma \to q\overline{q} \) splittings finite. We present our findings at the integrated and differential fiducial cross-section levels for the LHC Run III centre-of-mass energy of \( \sqrt{s} \) = 13.6 TeV. In addition, we quantify the impact of subleading NLO effects, in particular, electroweak Sudakov logarithms and non-resonant QCD backgrounds. Two analysis strategies are employed and compared, namely with and without the resonance-enhancing requirement on the invariant mass of the two light jets, |Mjj – mW| < \( {\mathcal{Q}}_{\mathrm{cut}} \) = 20 GeV, illustrating the relationship between QCD background suppression, off-shell effects, interferences, and complete NLO corrections.
