 panel, we fix $\alpha=0.1$ and vary the ratio of the coupling $g_5/g$ for the ground state. Let us observe that in the scalar case, there is a visible dependence on $\alpha$. This is different with respect to the case of the vector mediator model in ref.~\cite{vonHarling:2014kha} (black-dashed line in the left panel of figure~\ref{fig:fig_IR_1S}). In this latter case, the combination of the dissociation rate and bound-state decay is such, that the resulting ionization factor is $\alpha$-independent.\footnote{For the bound-state decay width we use $\Gamma_{\textrm{ann}}^{1S}=M \alpha^5/2$, that corresponds to the para-positronium. For this comparison, we take the decay width in eq.~\eqref{decay_width_nS} at leading order, namely at $\mathcal{O}(\alpha_5 \alpha^4)$.} Also, for the model under study,  we obtain larger values of $\mathcal{I}_{1S}$ with respect to the Abelian vector mediator, pointing to an earlier contribution of bound-state effects. Nevertheless, one has to bear in mind that the bound-state formation process mediated by a scalar mediator features an  $\alpha^2$ suppression~\cite{Petraki:2015hla}, and so the overall impact of bound-state effects on the DM energy density is a non-trivial combination of $\mathcal{I}_{n}$ and $\langle \sigma_{\textrm{bsf}} v_{\textrm{rel}} \rangle$ for different models.
 
 
 In the right panel of figure~\ref{fig:fig_IR_1S}, one may see how decreasing values of $\alpha_5$, that make the bound-state decay width in eq.~\eqref{decay_width_nS} smaller, but at the same time leaving unaffected the dissociation thermal width, determines a larger ionized population of bound states in the same temperature window. The dashed lines correspond to different states ($1S$, $2S$ and $2P_0$) for the same value of $g_5=g/5$. Excited states remain ionized for longer times. 
 \begin{figure}[t!]
    \centering
    \includegraphics[width=0.45\textwidth]{Alpha_Mass_Omega_g5_over5_Linear.pdf}
    \hspace{0.5 cm}
   \includegraphics[width=0.45\textwidth]{Summary_BSF_SOM_ex3.pdf}
    \caption{Left: Parameter space in the $(M,\alpha)$-plane that reproduces the observed DM energy density for $g_5=g/5$. The purple dotted curve accounts for the ground state only, whereas the purple-dashed curve corresponds to the inclusion of excited states. Right: Ratio between the DM energy density as obtained with and without bound-state effects. The parameter $\alpha$ is fixed as a function of the mass $M$ to account for the observed DM energy density when bound-state effects are considered. Dashed (dotted) lines do (not) include excited states. }
    \label{fig:num_Boltzmann}
\end{figure}
 
As a final summarizing result, we give the solution of the effective Boltzmann equation \eqref{Boltzmann_eq_eff}. As usual, we solve the evolution equation for the DM yield $Y_X=n_X/s$, where $s=2 \pi^2 h_{\textrm{eff}}T^3/45$ is the entropy density; we assume that the dark states are at the same temperature as the plasma of SM
particles, and we set $ h_{\textrm{eff}}$ and $ g_{\textrm{eff}}$ to the values from ref.~\cite{Laine:2015kra}. In the left plot of figure~\ref{fig:num_Boltzmann}, the curves reproduce the observed DM energy density $\Omega_{\textrm{DM}}h^2 = 0.1200 \pm 0.0012$~\cite{Aghanim:2018eyx} in the parameter space $(M,\alpha)$. Both the Sommerfeld enhancement and bound-state effects are more prominent for large values of $\alpha$ as expected. One may notice the large impact upon including the Sommerfeld enhancement, both for the $S$- and $P$-wave annihilations (cf.\  eq.~\eqref{ann_cross_Sommerfeld}), with respect to the free annihilation cross section (cf.\  eq.~\eqref{NR_hard_cross_section_S}). In contrast to that, the contributions of the bound-state formation and decays are moderately relevant, albeit still non-negligible.




In order to single out the bound-state effects, we consider the ratio between the DM energy density as obtained by including or omitting the bound-state formation and decays (namely the second term on the right-hand side of eq.~\eqref{Boltzmann_eq_eff}), in addition to the Sommerfeld enhancement for the pair annihilations. This is shown in the right plot of figure~\ref{fig:num_Boltzmann}. The ratio is smaller than unity because, when neglecting bound-state formation, a smaller population of DM particles is annihilated away, and a larger energy density is found. The formation of bound-states and their decays into light scalars act as an additional channel for depleting DM. 
We notice that a smaller pseudoscalar coupling makes the bound-state formation term more relevant, and this traces back to the different powers of $\alpha$ and $\alpha_5$ entering various cross sections.\footnote{For the case when only the ground state is considered, and looking at the $S$-wave annihilations only, the effective cross section is $\langle \sigma_{\textrm{eff}} v_{\textrm{rel}} \rangle \approx 2 \pi \alpha \alpha_5 \langle S(\zeta) \rangle /M^2 + \pi \alpha^4 \langle S_{\textrm{bsf}}(\zeta) \rangle \mathcal{R}_{1S} /M^2 $. Here, $\langle S_{\textrm{bsf}}(\zeta) \rangle$ can be read off from eqs.~ \eqref{bsf_matrix_element_T} and \eqref{bsf_1S}.} For the smaller ratio  between the pseudoscalar and scalar couplings we consider, $g_5=g/10$, and the largest mass compatible with the observed relic density for $\alpha_{\textrm{max}}=0.3$, namely $M=13\,\textrm{TeV}$, we find a 30\%-36\% effect on the predicted energy density if bound-state formation is not included. The dotted lines correspond to the inclusion of the ground state only, whereas upon adding the excited states ($nS$ and $nP$ states with $n=2,3$) we obtain the dashed lines. Bound states with $n>4$ induce a contribution smaller than the uncertainty in the observed DM energy density in our numerical implementation. 

Upon solving the Boltzmann equation \eqref{Boltzmann_eq_eff}, we start the integration at temperatures larger than those strictly admitted by the hierarchy of scales \eqref{hiearchy}, which implies that $T \ll M \alpha$ is not always satisfied. This would amount to deriving the bound-state formation and dissociation rate at such temperatures, in the so-called screening regime, cf.\  refs.~\cite{Brambilla:2008cx,Escobedo:2008sy} for QCD and QED, and refs.~\cite{Biondini:2018pwp,Binder:2018znk} for a DM application.\footnote{Pair annihilations would be affected in terms of the thermal potential at such temperatures, and the corresponding modifications to the wave functions from a plasma-modified Schrödinger equation, cf.\  refs.~\cite{Kim:2016kxt,Biondini:2017ufr,Biondini:2018pwp,Binder:2018znk,Kim:2019qix}. From an EFT perspective, one has to integrate out the temperature scale before obtaining a pNREFT, the latter finally attained when removing potential scalars.}  However, due to our assumption for the self-coupling of the scalar, and upon neglecting the explicit interactions in $\mathcal{L}_{\textrm{portal}}$, a thermal mass for the scalar mediator does not appear and, moreover, $2 \to 2$ scattering processes with the medium constituents mediated by a soft scalar are absent at least in the Hard Thermal Loop limit~\cite{Thoma:1994yw}. Despite of the fact that bound-state formation and corresponding decays are more effective at later temperature stages, further investigations on this aspect can be worthwhile, and are left for future research on the subject. 
 
 
\section{Conclusions}
\label{sec:concl}
In this paper we have taken the first step towards the generalization of a potential non-relativistic effective theory for scalar mediators at finite temperature. In addition to the dynamically generated energy scales typical of bound states, there are also thermodynamic scales at play. Most notably, these are the plasma temperature and thermal masses. Our main goal is to contribute to the recent effort and the development of theoretical tools for estimating the near-threshold and bound-state effects on DM freeze-out calculations. In this scenario DM particles are indeed non-relativistic and slowly moving in the early universe plasma in the relevant temperature window. This