 QCD is shown for $\mu_0=H_T/2$ and $\mu_0=\mu_{fix}$. Results
are obtained with the default PDF set NNPDF3.1. The full off-shell calculation, NWA and $\textrm{NWA}_{\textrm{LOdec}}$ results are presented with the corresponding Monte Carlo error and a theoretical uncertainty estimate obtained  through scale variation. We note here, that in the full off-shell calculation the scale uncertainties are calculated with the $7$-point scale variation while for NWA and $\textrm{NWA}_{\textrm{LOdec}}$ the scale uncertainties are obtained with the $3$-point scale variation. The latter approach, however, leads to similar scale uncertainties as the former one, as discussed in section \ref{sec:tth-int}. At LO, the deviations between the NWA and the full off-shell calculation are about $0.5\%$ for $\mu_0=H_T/2$ and $0.3\%$ for $\mu_0=\mu_{fix}$. Thus, they are negligible compared to the large scale uncertainties of about $30\%$. At NLO, the off-shell effects are about $0.4\%$ for both scale settings and, again, significantly smaller than the scale uncertainties of about $5\%$. These differences are consistent with the expected error of the NWA that is of the order of $\Gamma_t/m_t\approx 0.8\%$ \cite{Fadin:1993kt}. We notice that the maximum of the scale variation for both scales is essentially reproduced in the NWA and that the differences between the full off-shell calculation and the NWA do not exceed $0.5\%$ for all scale choices required in the $3$-point scale variation. The integrated fiducial cross section in the $\textrm{NWA}_{\textrm{LOdec}}$ is about $5\%$ $(6\%)$ larger for $\mu_0=H_T/2$ ($\mu_0=\mu_{fix}$) than the result in the NWA. Thus, the inclusion of QCD corrections in the top-quark decays leads to a reduction of the integrated cross section. Moreover, the scale uncertainties increase to about $9\%$ when the NLO QCD corrections in the top-quark decays are omitted.
\begin{figure}[t!]
	\begin{center}
		\includegraphics[width=0.49\textwidth]{diff_nwa_mbe.pdf}
		\includegraphics[width=0.49\textwidth]{diff_nwa_ht_tilde_long.pdf}
		\includegraphics[width=0.49\textwidth]{diff_nwa_mbb_long.pdf}
		\includegraphics[width=0.49\textwidth]{diff_nwa_ptbb_long.pdf}
	\end{center}
	\caption{\label{fig:diff-nwa1} \it
		Differential distributions at NLO QCD for the observables $M(be^+)_{min}$, $H_T$, $M_{b_1b_2}$ and $p_{T,\,b_1b_2}$ for the $pp\to e^+\nu_e\mu^-\bar{\nu}_{\mu}b\bar{b}\,H$ process at the LHC with $\sqrt{s}=13\textrm{ TeV}$. Results are presented for $\mu_0=H_T/2$. The NNPDF3.1 PDF set is employed. The upper panels show full off-shell, NWA and $\textrm{NWA}_{\textrm{LOdec}}$ results. We additionally provide theoretical uncertainties as obtained from the scale dependence for the full off-shell case. Also given are Monte Carlo integration errors. The lower panels display the ratios of the two NWA results to the full off-shell result. }
\end{figure}

Even though full off-shell effects of the top quarks and $W$ gauge bosons are formally suppressed by the corresponding widths and are small for the inclusive cross section, they can be strongly enhanced in exclusive observables. Specifically, for dimensionful observables at the differential level, the contribution from single- and non-resonant Feynman diagrams can be enhanced and even become larger than the contribution from double resonant diagrams in certain phase-space regions. The latter comprise high $p_T$ regions and kinematical edges, that are  related to the masses of the on-shell particles and the cuts that all final states have to pass. As an example, in Figure \ref{fig:diff-nwa1}, we show differential distribution at NLO in QCD for the observables $M(be^+)_{min}$, $H_T$, $M_{b_1b_2}$ and $p_{T,b_1b_2}$. We provide theoretical results with the full off-shell effects included as well as predictions in the NWA and $\textrm{NWA}_{\textrm{LOdec}}$. We utilize $\mu_0=H_T/2$ and the NNPDF3.1 PDF set. The lower panels display the ratios of the corresponding NWA result to the full off-shell one. We additionally provide theoretical uncertainties as obtained from the scale dependence for the full off-shell case. The observable $M(be^+)_{min}$ is defined as the minimum mass of the positron and a $b$-jet. At LO in the NWA, this observable is limited by $\sqrt{m_t^2-m_W^2} \approx 153$ GeV due to the finite mass of the top quarks and $W$-bosons. At NLO, this kinematical edge is smeared out by real radiation, but the NWA is still unable to recover the full calculation in the vicinity of $ M(be^+)_{min} \approx 153$ GeV. For  $M(be^+)_{min} \le 153$ GeV, the NWA leads to good agreement with the off-shell calculation and the differences between the two cases are significantly smaller than the scale uncertainties. The missing corrections in the decay of the top quark lead to further shape distortions in these regions of the phase space, which, this time however, are larger than the scale variation of the full result. For $M(be^+)_{min} \ge 153$ GeV, on the other hand, we obtain substantial non-factorizable corrections that are of the order of $50\%-60\%$. For $H_T$, we observe that the differences between the NWA and the off-shell calculation are much smaller than the scale uncertainties at the beginning of the spectrum. Towards the tails, these differences rise up to about $10\%$, which is roughly the size of the corresponding scale uncertainties. In the tails, NWA and $\textrm{NWA}_{\textrm{LOdec}}$ are very similar, while for $H_T \approx 500$ GeV we find additional shape distortions for the latter approach, close to $10\%$. For $M_{b_1b_2}$ and $p_{T,\,b_1b_2}$ we once more observe that the NWA and the off-shell calculation agree well for small and moderate values. However, in the tails, the differences are equal to the scale uncertainties of about $10\%$ for $M_{b_1b_2}$ and $20\%$ for $p_{T,\,b_1b_2}$. Also here the $\textrm{NWA}_{\textrm{LOdec}}$ leads to additional shape distortions of about $10\%$. 
\begin{figure}[t!]
	\begin{center}
		\includegraphics[width=0.49\textwidth]{diff_nwa_phibb.pdf}
		\includegraphics[width=0.49\textwidth]{diff_nwa_ye.pdf}
	\end{center}
	\caption{\label{fig:diff-nwa2} \it
		Differential distributions at NLO QCD for the observables $\Delta\phi_{b_1b_2}$ and $y_{e^+}$ for the $pp\to e^+\nu_e\mu^-\bar{\nu}_{\mu}b\bar{b}\,H$ process at the LHC with $\sqrt{s}=13\textrm{ TeV}$. Results are presented  for $\mu_0=H_T/2$. The NNPDF3.1 PDF set is employed. The upper panels show full off-shell, NWA and $\textrm{NWA}_{\textrm{LOdec}}$ results. We additionally provide theoretical uncertainties as obtained from the scale dependence for the full off-shell case. Also given are Monte Carlo integration errors. The lower panels display the ratios of the two NWA results to the full off-shell result.}
\end{figure}

In Figure \ref{fig:diff-nwa2}, we exhibit differential distributions for the two dimensionless observables $\Delta\phi_{b_1b_2}$ and $y_{e^+}$. For $\Delta\phi_{b_1b_2}$, we find that the NWA is able to recover the full off-shell calculation over the entire range. Thus, full off-shell effects are negligible compared to scale uncertainties. On the other hand, $\textrm{NWA}_{\textrm{LOdec}}$ leads to shape distortions in the range of $2\%-10\%$. The latter cannot be simply corrected by a re-scaling factor. Likewise, for the rapidity of the positron, $y_{e^+}$, we find that full off-shell effects are rather insignificant. Furthermore, the ${\rm NWA}_{\rm LOdec}$ approach introduces an overall normalisation correction of the