${\cal CP}$ structure of the top-quark Yukawa interaction: NLO QCD corrections and off-shell effects

Since its discovery at the Large Hadron Collider in 2012 the Higgs boson has arguably become the most famous of the Standard Model particles and many measurements have been performed in order to assess its properties. Among others, these include measurements of the Higgs boson's ${\cal CP}$ state which is predicted to be ${\cal CP}$-even. Even though a pure ${\cal CP}$-odd state has been ruled out, a possible admixture of a ${\cal CP}$-odd Higgs state has yet to be excluded. In this work we present predictions for the associated production of a leptonically decaying top quark pair and a stable Higgs boson $p p \to e^+ \nu_e\, \mu^- \bar{\nu}_\mu\,b\bar{b}\,H$ with possible mixing between ${\cal CP}$-even and ${\cal CP}$-odd states at NLO in QCD for the LHC with $\sqrt{s} = 13$ TeV. Finite top-quark and gauge boson width effects as well as all double-, single- and non-resonant Feynman diagrams including their interference effects are taken into account. We compare the behaviour of the ${\cal CP}$-even, -odd and -mixed scenarios for the integrated fiducial cross-sections as well as several key differential distributions. In addition, we show that both NLO corrections and off-shell effects play an important role even at the level of integrated fiducial cross-sections and that these are further enhanced in differential distributions. Even though we focus here on the Standard Model Higgs boson, the calculations could be straightforwardly applied to models that have an extended Higgs-boson sector and predict the existence of ${\cal CP}$-odd Higgs-like particles, such as the two-Higgs-doublet model.