m_t$, annihilations to top pairs become kinematically allowed and $\sigv$ receives a contribution from the operator proportional to $d_4$. In this case, we can once again generate the correct relic abundance, even when $\lambda_{\xi h} \ll \lambda_{\xi h}^{\rm NC}$. This corresponds to all the points at large masses with small values of $\sigma_{\rm DMp}$ and $a_{d1}$ in the LH side plot of figure~\ref{fig:sDD}. Notice though that this feature is not as strongly present in the right hand (RH) side plot of figure~\ref{fig:sDD} for $F=2.5$\,TeV\@. This is simply due to the fact that for such large $F$ values and $m_{\rm DM} > m_t$, non-derivative operators, such as the one proportional to $d_4$, are always subdominant, as already anticipated by the RH side plot of figure~\ref{fig:ops}. Finally, we also emphasize that for $F=2.5$\,TeV, $\sigma_{\rm DMp}$ only deviates significantly from its non-composite value for higher DM masses ($m_{\rm DM} \gtrsim 150$\,GeV), as expected from eq.~(\ref{sDDapp}) and in agreement with the discussion in section~\ref{effects}. From figure~\ref{fig:sDD} we already see that for $m_{\rm DM}> 200$\,GeV to be consistent with LUX we need $\vert a_{d1}\vert \lsim 0.2$ for $F = 0.8$\,TeV, higher values are however possible if we increase the scale of compositeness. ]]>

500$\,GeV, even before applying the direct, indirect and Higgs experimental constraints. Once $m_{\rm DM} > m_h$, the annihilation into two Higgses can contribute significantly to $\sigv$, which can be large even if $\lambda_{\xi h}^{DD}\simeq 0$ due to derivative coupling between DM and Higgs~\cite{Frigerio-ml-2012uc}. As a result, the annihilation cross-section becomes too large, suppressing the relic abundance below the WMAP value. However this only happens for small enough values of $F$, as seen in figure~\ref{fig:O6}. ]]>

\frac{1}{16 \pi m_{\rm DM}^2} |\mathcal{M}_T|^2 \simeq \frac{g_V^2}{32 \pi m_{\rm DM}^2}\, , \end{equation} resulting in too small relic abundances, unless $m_{\rm DM} \gtrsim 500$\,GeV\@. This is a well known result from the IDM~\cite{LopezHonorez-ml-2006gr,Hambye-ml-2009pw} and it still holds in the composite case, since the contributions from the dimension-6 operators to $\mathcal{M}_T$ are negligible. Notice that, in contrast with the non-composite case~\cite{Hambye-ml-2009pw}, for sufficiently large $m_{\rm DM}$ the annihilation cross-section is driven by the term proportional to $a_{d2}$. Let us stress though that, as mentioned in section~\ref{compDM}, our effective Lagrangian approach is only valid for $m_{\rm DM} \lesssim F$. From eq.~(\ref{ampL}) we see that if the $\mathcal{M}_L$ contribution had to be suppressed one would need small mass splittings among the components of the DM doublet ($\Delta m_{\xi}^2/\bar{m}_{\rm DM}^2 \ll 1$) and the combination \begin{equation} \lambda_{\xi h}^{RD} \frac{v^2}{m_{\rm DM}^2} + 2 a_{d2} \frac{v^2}{F^2} \label{condLeff} \end{equation} should be small. Since $|\lambda_{\xi h}^{RD}|$ can not be made arbitrarily large (otherwise $H^0 H^0 \to h h$ enhances $\sigv$), we will see below that the above condition limits the possible values~of~$a_{d2}$. ]]>

m_h$. \begin{figure}[t] \centering \includegraphics[width=7.5cm]{doubletFIXED_800_sDD_channels.png}\hfill \includegraphics[width=7.5cm]{doubletFIXED_2500_sDD_channels.png}\\ \includegraphics[width=7.5cm]{doubletFIXED_Inf_sDD_channels.png}\hfill \includegraphics[width=7.5cm]{doubletFIXED_HighMass.png} ]]>

10^{-2}$) lying between the resonance region and the $W$ threshold for large values of $F$. Therefore, most of the low mass region of generic composite doublet DM models is already experimentally excluded or theoretically disfavored. In contrast, the high mass region has just been marginally probed by LUX and it even Xenon1T will not be able to test it in its entirety. \begin{figure}[t] \centering \includegraphics[width=7.5cm]{doubletFIXED_800_lbar2_constraints.png}\hfill \includegraphics[width=7.5cm]{doubletFIXED_2500_lbar2_constraints.png} ]]>

10^{-2}$. Therefore here we only discuss the high mass ($m_{\rm DM} > 500$\,GeV) solutions. One important consequence from specifying the $\mG/\mH$ coset is that it fixes the value of $a_{d2}$. As we have seen above, the high mass region requires either large values of $F$ or small values of $a_{d2}$ (see figure~\ref{fig:doubletad2}). Hence, for $\mG/\mH = \SO(6)/\SO(4)\times \SO(2)$, where $a_{d2} = 1$, we do not expect viable solutions for large DM masses and small values of $F$. This is highlighted in the LH of figure~\ref{fig:fscan}, where we allow $\Omega_{\rm DM} h^2$ to vary outside the $0.09 \lesssim \Omega_{\rm DM} h ^2 \lesssim 0.12$ range and show the relic density as a function of $m_{\rm DM}$ for different $F$ values, shown by the color gradient. For this plot, we take $m_{A^0} = m_{H^\pm} = m_{\rm DM}$ and $\bar{\lambda} = \lambda_i' = 0$ since this choice approximately maximizes (minimize) the relic abundance ($\sigv$) and we take $d_4=d_6=0$, since these coefficients play no major role in the high mass region. We can understand the behavior of $\Omega_{\rm DM} h ^2$ in the LH of figure~\ref{fig:fscan} by analyzing the $m_{\rm DM}$ and $F$ dependence of $\sigv(\xi\xi\rightarrow VV)$ derived in eq.~(\ref{ampL}), but in the ``pure gauge limit'' (all quartic couplings set to zero). The relic abundance first grows with $m_{\rm DM}$ due to the usual dependence $\Omega_{\rm DM} h ^2\propto 1/\sigv\propto m_{\rm DM}^2$. This is indeed what we observe for the lowest range of masses in figure~\ref{fig:fscan} (LH side). The non-composite case in the same limit is plotted for reference with gray points and also show this growing dependence in $m_{\rm DM}$. When $m_{\rm DM}$ becomes comparable to $F$, the dimension-6 derivative operator starts to dominate $\sigv$ and the contribution from the $a_{d2}$ term results in $\Omega_{\rm DM} h ^2\propto 1/\sigv\propto 1/m_{\rm DM}^2$. For a given $F$ scale there is thus a fixed mass range, bounded from below and above, of viable DM candidates, i.e.\ in agreement with the CMB bound depicted with vertical dashed gray lines in figure~\ref{fig:fscan} (LH side). This viable mass range widen with increasing $F$. Indeed, we see that for e.g.\ $F = 2.4$\,TeV we have 550\,GeV $\lesssim m_{\rm DM} \lesssim 1.2$\,TeV, while for e.g.\ $F= 2.1$\,TeV we have 600\,GeV $\lesssim m_{\rm DM} \lesssim 800$\,GeV\@. In addition, once $F < 2$\,TeV, $\Omega_{\rm DM} h^2 < 0.09$ for any value of $m_{\rm DM}$ in the large mass range. Therefore, for viable composite scenarios with $\mG/\mH = \SO(6)/\SO(4)\times \SO(2)$ one needs to consider $F > 2$\,TeV\@. In figure~\ref{fig:fscan} (RH side) we show results for a scan on the parameter space of $\mG/\mH = \SO(6)/\SO(4)\times \SO(2)$, where the non-derivative coefficients are still free parameters. We have explicitly verified that even when all non-derivative parameters are allowed to vary there are no viable solutions for $F = 0.8$\,TeV\@. We thus present points only for $F = 2.5$\,TeV and $F\rightarrow \infty$. As shown by the results, for $F = 2.5$\,TeV, we only obtain the correct relic abundance for 500\,GeV $\lesssim m_{\rm DM} \lesssim 1.1$ TeV, as expected from our discussion above. \begin{figure}[t] \centering \includegraphics[width=7.5cm]{doubletSO6F_Fscan.png}\hfill \includegraphics[width=7.5cm]{doubletSO6_HighMass.png} ]]>

2$ of \SU(2)$_L$ we first focus on the case of the triplet. Later we will discuss how these results generalize to larger representations. In the triplet case, we use the following generators (in the spherical basis): \begin{equation} \label{eq:generatortripletspheric} \Gamma_1=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & -1 & 0 \\ -1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} \, , \quad \Gamma_2=\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & i & 0 \\ -i & 0 & -i \\ 0 & i & 0 \end{pmatrix} \, , \quad \Gamma_3=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}\,,\\ \end{equation} and $\xi=(T^+, T^0,T^-)^T$. Using eq.~(\ref{tripLeff0}) with the above representation for the $\Gamma$ matrices, we obtain the following scalar mass spectrum: \begin{align} m^2_{\rm DM} \equiv m^2_{T^0}&=\mu_{\xi}^2 +v^2 \left(\frac12 \lambda_3+\frac14 \lambda_3 \lambda_3'\frac{v^2}{F^2}+ \lambda_5\frac{v^2}{F^2}\right) \, , \label{eq:mT0}\\ m^2_{T^\pm}&=\mu_{\xi}^2 +v^2 \left(\frac12 \lambda_3+\frac14 \lambda_3 \lambda_3'\frac{v^2}{F^2}+ \frac12 (\lambda_4+\lambda_5)\frac{v^2}{F^2}\right) . \label{eq:mTc} \end{align} Notice that in the triplet case, once the values of $m_{\rm DM}$, $\lambda_4$ and $\lambda_5$ are given, the value of $m_{T^\pm }$ is fixed. In the non-composite case ($F \to \infty$), the triplet is exactly degenerate (except for small EW loop corrections, see~\cite{Cirelli-ml-2005uq}). In the latter case, either LEP constraints excluding the existence of low mass new charged particles or too strong annihilation into gauge (virtual) bosons\footnote{In the triplet case $T^0 T^0 \to V V^{*}$ is already dominant for $m_{\rm DM} \gtrsim 50$\,GeV.} prevent the existence of viable DM candidate in the low mass range (i.e.\ for $m_{\rm DM} 70$\,GeV, if $F \lesssim 1.5$\,TeV\@. Larger values of $F$, however, will not generate sufficient mass splitting to allow for viable solutions in the low mass region. We will analyze this in more details in section~\ref{sec:scan-results-triplet}. The effective Higgs-DM coupling in the triplet case is simply: \begin{align} \lambda_{\xi h}& = \frac{\bar{\lambda}}{2} - a_{d1}\frac{p_h^2}{4 F^2}\; ,\label{eq:lxihT}\\ \mbox{where } \bar{\lambda} &= \lambda_3 \left(1 + \lambda_3'\frac{v^2}{F^2}\right) + 4 \lambda_5 \frac{v^2}{F^2}\, .\label{eq:lbarT} \end{align} Unlike the doublet case, the DM-gauge boson couplings are only affected by one single coefficient ($a_{d4}$) in eq.~(\ref{tripLeff0}). While the contributions to the quartic vertices ($\xi^2 V_\mu V^\mu$) are usually negligible: \begin{equation} \mathcal{L}_{\rm triplet} \supset 2 g^2 \left(1 - a_{d4} \frac{v^2}{F^2}\right) T^0 T^0 W^+ W^- \, , \end{equation} as they are always suppressed by $v^2/F^2$, the triple vertex couplings ($\xi^2 V_\mu$) can be enhanced at large $m_{\rm DM}$. In addition let us emphasize that, for the triplet and higher representations, the coupling to the gauge bosons is always larger than in the doublet case because it scales like $(n^2-1)^2$. In the case of the doublet, we saw that once $m_{\rm DM} \simeq m_W$, the (co-)annihilation into gauge bosons generate too large $\sigv$, resulting in too low relic densities until $m_{\rm DM} \gtrsim 500$\,GeV\@. This feature is only enhanced in the triplet case, since its couplings to gauge bosons are larger, as it is well known in the non-composite case~\cite{Cirelli-ml-2005uq,Hambye-ml-2009pw}. We also point out that, due to $CP$-invariance, $Z$ decays to DM pairs are always forbidden, so the $Z$-width constraints do not apply to the case of real triplets. ]]>

10^{-2}$ are still viable and will eventually be tested by Xenon1T experiment. We conclude that, despite allowing for large mass splittings, the composite triplet scenario, alike the non-composite case, remains mostly excluded in the low mass region. In a way analogous to the doublet scenario, for $m_{\rm DM} \gg m_W$, the annihilation cross-section into gauge bosons becomes suppressed enough to give rise to the right relic abundance. In this high mass region, it takes the following form: \begin{align} \sigv|_{T_0T_0\rightarrow VV} & \simeq \frac{1}{32 \pi m_{\rm DM}^2}\left(2 |\mathcal{M}_T|^2 + |\mathcal{M}_L|^2 \right) \mbox{, where}\nonumber\\ \mathcal{M}_{T}\left(T^0 T^0 \to V_T V_T \right) & \simeq 2 g_V^2 \left[ 1 - a_{d4}\frac{v^2}{F^2} + \frac{\bar{\lambda}}{8} \frac{v^2}{m_{\rm DM}^2} - a_{d1} \frac{v^2}{2 F^2} m_{\rm DM}^2 \right] \nonumber \\ & \simeq 2 g_V^2 \left( 1 - a_{d4}\frac{v^2}{F^2} + \frac{v^2}{2 m_{\rm DM}^2}\lambda_{\xi h}\right)\; , \nonumber \\ \mathcal{M}_{L}\left(T^0 T^0 \to V_L V_L \right) & \simeq \frac{g_V^2}{2}\frac{v^2}{m_V^2} \lambda_{\xi h} \simeq \lambda_{\xi h} \, , \label{ampTtrip} \end{align} where we have taken $m_{\rm DM} \simeq m_{T^+}$, since the small mass splitting can be neglected in the high mass region. Notice that in the large mass regime, we typically have $\lambda_{\xi h} v^2/(2 m_{\rm DM}^2)\ll 1$, and for the $F \gg v$ considered here we also have $a_{d4} v^2/F^2 < 1$. These two contributions to ${\cal M}_T$ will thus typically be small. \begin{figure}[t] \centering \includegraphics[width=7.5cm]{triplet_HighMass.png} \includegraphics[width=7.5cm]{triplet_2500_ad1.png} ]]>

m_{\rm DM} \gtrsim 1.7$\,TeV\@. Hence there are no values of $F$ for which there are solutions in \emph{both} the low mass and the high mass regions. Furthermore, since the low mass region is already excluded by LUX, the triplet case can only provide a DM candidate for $F > 1.7$\,TeV\@. ]]>