Multi-scale improved predictions for $\boldsymbol{pp \to t\bar{t}W^+ +X}$

We compare standard fixed-order NLO QCD predictions for the full off-shell $pp \to e^+\nu_e\, \mu^-\bar{\nu}_\mu\, \tau^+\nu_\tau\, b\bar{b} \, (j) + X$ processes in the multi-lepton decay channel with the results obtained using the MiNLO approach. The MiNLO method, now also implemented in the Helac-NLO framework, provides a dynamic determination of the renormalization and factorization scale settings by identifying the most likely branching histories of the additional jets through an inverse $k_T$-clustering algorithm. The inclusion of Sudakov form factors accounts for the large logarithms that arise in the presence of widely separated scales. We perform a dedicated comparison of the two approaches at both the integrated and differential (fiducial) cross-section levels for the LHC Run II energy of $\sqrt{s}=13$ TeV. Finally, we present the merging of the full off-shell predictions for $pp \to e^+\nu_e\, \mu^-\bar{\nu}_\mu\, \tau^+\nu_\tau\, b\bar{b} \, + X$, $pp \to e^+\nu_e\, \mu^-\bar{\nu}_\mu\, \tau^+\nu_\tau\, b\bar{b} \, j +X$ and $pp \to e^+\nu_e\, \mu^-\bar{\nu}_\mu\, \tau^+\nu_\tau\, b\bar{b} \, jj$, with the aim of improving the description of the underlying $pp \to e^+\nu_e\, \mu^-\bar{\nu}_\mu\, \tau^+\nu_\tau\, b\bar{b} + X$ process.