We propose a simplified model of dark matter with a scalar mediator to
accommodate the di-photon excess recently observed by the ATLAS and CMS
collaborations. Decays of the resonance into dark matter can easily account
for a relatively large width of the scalar resonance, while the magnitude of
the total width combined with the constraint on dark matter relic density
leads to sharp predictions on the parameters of the Dark Sector. Under the
assumption of a rather large width, the model predicts a signal consistent with

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2\text{\,fb}$ at $\sqrt{s}=13$\,TeV, for different choices of the DM mass $M_\psi=(350,300,200,100)\,\text{GeV}$, indicated on the boundaries of the regions. For the purpose of illustration, we fixed $g_{\rm DM} = 2.7$ and $\Lambda = 10 \tev$. The red dashed contours correspond to the production cross section at 13\,TeV for a scalar singlet $\sigma(pp\to S)$ in pb. ]]>

2\text{\,fb}$ depends on the value of the DM mass $M_\psi$ once the coupling strength $g_{\rm DM}$ is fixed to be $\mathcal{O}(1)$. At fixed dark matter mass, the boundary of the viable signal strength region reproduces the expected parametric dependence on $g_{GG}^2 g_{BB}^2$. The maximum signal strength is achieved when the DM mass approaches the kinematical limit $M_\psi\sim\frac{m_S}{2}$ and the invisible width controlled by~\eqref{invisibleCPeven} is reduced. From figure~\ref{BRplot} and figure~\ref{xsecplot} we see an interesting tension between enhancing the $\gamma \gamma$ signal strength and the total width at the same time. A large cross section in $\gamma\gamma$ would prefer a DM mass close to the kinematic threshold in order to suppress $\sigma(S\to \mathrm{invisible})$. On the other hand, a large width of order $\frac{\Gamma_{\rm tot}}{m_S} \sim \text{few} \%$ prefers a DM mass of $O(100)$\,GeV\@. In the following we will see how these two constraints together with the LHC-8TeV bounds select a specific region of the $g_{GG}$-$g_{BB}$ plane where also a viable DM candidate can be accommodated. \paragraph{Dark matter relic abundance:} the model we discuss also aims to account for the observed relic abundance of DM in the Universe, i.e.\ $\Omega h^2 \simeq 0.12$. The annihilation cross section of DM into SM particles is driven by higher dimensional operators, typically resulting in annihilation rates which are too low to obtain the correct $\Omega h^2$ for generic values of the dimensionless couplings and dark matter mass. The correct value for $\Omega h^2$ can be obtained if the annihilation is kinematically enhanced, i.e.\ if the mass of the DM is ``close''\footnote{While the term ``close to the resonance'' is often used in the context of resonant dark matter annihilation, it is seldom pointed out that the relevant measure for correct relic density is in fact $|m_S - 2M_\Psi| / \Gamma_{\rm tot} \lesssim O(1)$.} to the singlet resonance. Note that this is exactly the same region where the signal strength in $\gamma\gamma$ is maximized as shown in figure~\ref{xsecplot}. In the same region the kinematical suppression reduces the invisible width of the scalar~\eqref{invisibleCPeven} and hence the total width (see figure~\ref{BRplot}). We then expect the Dark Matter mass and coupling $g_{\rm DM}$ to be fully determined by the intersection of the relic density and resonance width constraints. Indeed, requiring a large width in combination with relic density alone provides strong constraints on the parameters of the Dark Sector. In order to clarify this aspect further, we performed a Markov-chain exploration of the model parameter space in $\{ g_{GG}, g_{BB}, g_{\rm DM},$\linebreak $M_{\psi} \}$. Figure~\ref{fig:markov} shows the results, where we projected the four dimensional parameter space onto the $(g_{\rm DM}, M_{\psi})$ plane. The correct relic density and a large total width can be obtained only in the region of dark matter mass of the order of $M_{\psi} \sim 300$\,GeV, regardless of the values of $g_{BB}$ and $g_{GG}$. In the following we will show that some of the model points which satisfy both the relic density and the large width requirement (red diamonds in figure~\ref{fig:markov}) are also able to accommodate the observed di-photon signal strength. The exact desired values of the Dark Sector will depend on the value of the total width and somewhat on the spin of the DM particle and the chiral nature of the di-photon resonance. However, the overall conclusion that the requirement of a total width combined with relic density will essentially fix the parameters of the Dark Sector appears to be robust and weakly dependent on the remaining model parameters. \begin{figure} \centering \includegraphics[width=0.6\textwidth]{./my_figs/markov_mpsi_gdm_width_3_to_9.pdf} ]]>

500 \gev$. \item Recent CMS di-jet searches for resonances at $\sqrt{s} = 8 \tev$~\cite{Khachatryan-ml-2015sja} provide weak limits for production cross section of $\sigma(jj) \lesssim 1 $ pb for scalar resonances which couple dominantly to $gg$ of mass around the TeV scale. We will adopt this limit for a scalar resonance of $m_S \sim 750 \gev$ as a conservative estimate. \item The ATLAS search~\cite{Aad-ml-2015kna} provides a bound on the $ZZ$ cross section of the order $\sigma_{ZZ}~<~12$\,fb for a scalar resonance of $m_S \sim 750$\,GeV\@. In our scenario the $ZZ$ cross section is suppressed with respect to the $\gamma \gamma$ cross section by a factor $( \frac{s_W}{c_W} )^4 \sim 0.1$, since we fixed $g_{WW}\approx0$. Hence the bound on $\sigma_{ZZ}$ is less relevant than the ones on $\sigma_{\gamma\gamma}$ and $\sigma_{Z\gamma}$. \end{itemize} \paragraph{Dark matter detection constraints:} beside collider bounds, we expect that our dark matter model can also be constrained by direct and indirect detection experiments. Direct detection experiments can constrain the model since the lagrangian~(\ref{lagrangian}) induces the following effective operator between the dirac dark matter and the gluons \be \mathcal{L}_{\rm eff} \supset \frac{ g_{\rm DM} g_{GG}}{\Lambda m_S^2} \bar \psi \psi G_{\mu \nu} G^{\mu \nu} . \ee Notice that the strength of this operator is correlated with the requirement on the large total width as shown in figure~\ref{fig:sexy}. The resulting spin independent cross section for DM scattering off nucleons is then given by (see e.g.~\cite{Chu-ml-2012qy}) \be \label{DDsigma} \sigma_{SI}^{(p,n)} = \frac{1}{\pi} \frac{(m_{\chi} m_N)^2}{(m_{\chi}+m_N)^2} \left( m_{p,n}\frac{g_{\rm DM}g_{GG}}{\Lambda m_S^2} f^{(p,n)}_G \right)^2, \ee where $f^{(p,n)}_G=\frac{8 \pi}{9 \alpha_s} \left( 1-\sum_{q=u,d,s} f^{(p,n)}_q \right)$ is the gluon form factor and $\alpha_s$ is evaluated at the scale of the singlet mass. In our estimate of the direct detection constraints we are neglecting subleading operators which will be generated by the running from the UV scale to the typical scale of direct detection experiment $(\approx \text{GeV})$. This operators should be added in a more precise treatment of direct detection bounds.\footnote{We thank Paolo Panci for interesting discussions on this point.} The LUX experiment~\cite{Akerib-ml-2013tjd} provides a limit on the contact interaction between scalar mediators and gluons of $\sigma_{SI} \lesssim 4 \times 10^{-45}$ cm$^2$ for a dark matter of mass around $300$\,GeV. Concerning indirect detection, the annihilation is velocity suppressed in the case of a real scalar mediator. We hence do not expect strong bounds on our model from measurements of galactic gamma ray fluxes. We regardless estimate the cross section for annihilation of galactic DM into photons in our benchmark points for completeness. Recent measurements of galactic gamma rays from the FERMI collaboration~\cite{Ackermann-ml-2015lka} put a bound of $\langle \sigma v \rangle_{\gamma \gamma} \lesssim 10^{-28} \frac{\text{cm}^3}{\text{s}}$ for a DM mass of $O(300)$\,GeV that we adopt for our scenario.\footnote{The bound depends on the halo profile and varies in the range $(10^{-27}$--$10^{-28}) \frac{\text{cm}^3}{\text{s}}$.} \paragraph{Combined constraints:} \begin{figure} \centering \includegraphics[width=0.6\textwidth,trim = 0 0 0 25,clip]{./my_figs/final_max_min_v2_m.pdf} ]]>

2 $\,fb at LHC13TeV\@. We marginalize over $M_{\psi}=[25,600]\,\text{GeV}$ and $g_{\rm DM}=[0.1,3]$, where we always require $\Omega h ^2 \leq 0.12$. Regions above dashed and dotted lines are ruled out by individual searches specified on the plot, where we use dotted lines to represent the weakest limits in the marginalization and the dashed lines for the strongest limits. The solid blue line and the shaded region below it corresponds to the region of parameter space which can not account for a large width of the di-photon resonance. The points labeled as capital $P_{1-4}$ represent the benchmark model points in $(g_{GG}, g_{BB}$) of~(\ref{benchmarks}), we use as illustrations in the paper. The direct detection bounds labeled $DD$ assume $\Omega h ^2 = 0.12$. ]]>

2$\,fb and non over-abundance of dark matter. Figure~\ref{xsecplot2} shows the results, where we projected the four dimensional scan onto the $(g_{GG}, g_{BB})$ plane, marginalizing over $g_{\rm DM}$ and $M_{\psi}$. Signal yield of $\sigma_{\gamma\gamma} > 2 $\,fb can be obtained only in the region above and to the right of the solid blue line (as expected since the $\gamma\gamma$ cross section scales like $g_{GG}^2 g_{BB}^2$ for a fixed total width). In the same plot we display the bounds from LHC-8TeV searches as dashed/dotted lines, where dashed lines represent the strongest limits in the marginalization and dotted lines stand for the weakest limits. Requiring $\Omega h^2 = 0.12$ via thermal annihilation of $\psi$ will fix the value of $M_{\psi}$ and $g_{\rm DM}$, as shown in figure~\ref{fig:markov}. The resulting limits then sit between the dashed and dotted lines, as we will illustrate for the choice of benchmark points $p_{1\ldots 4}$. The direct detection limits displayed in figure~\ref{xsecplot2} assume that the local dark matter density corresponds always to $\Omega h^2 \approx 0.12$, although the relic density contribution from thermal annihilation of $\psi$ does not have to account for all of the observed relic density. The direct detection limits could hence be weaker than the ones we present. However, note that even in most stringent case of direct detection limits, the searches for MET+$j$ provide the strongest absolute constraints on $g_{GG}$. Hence, direct detection results do not provide relevant limits on $g_{GG}$ once MET+$j$ constraints are taken into account. The other 8TeV collider results are instead able to constrain the combination of $g_{BB}$ and $g_{GG}$. Our results show that the non-shaded portion of parameter space with $g_{GG} \lesssim 0.3$, assuming $g_{BB} \sim O(1)$, is still allowed by the LHC-8TeV data as well as by dark matter direct detection constraints. Note that this region of model parameters is also able to accommodate the di-photon excess signal strength. \paragraph{Benchmark points:} table~\ref{tab:constraints} shows a summary of all the experimental constraints on our scenario for the four benchmark model points in table~\ref{tab:fit}. Benchmark point $2$, with $(g_{GG},g_{BB})=(0.25,2)$, gives the largest yield in the di-photon signal (see table~\ref{tab:fit}) and it is already severely constrained by the $\gamma \gamma$ final state. Interestingly, requiring the correct DM relic abundance for that choice of $g_{GG}$ and $g_{BB}$, and hence fixing $g_{\rm DM}$ and $M_{\psi}$ to the values in table~\ref{tab:fit}, enhances the $Z\gamma$ branching ratio making the benchmark 2 also excluded by $Z\gamma$ searches at LHC-8TeV. The other benchmark points are all within the allowed experimental bounds, both from collider and from dark matter experiments, and can provide viable scenarios to accommodate the di-photon excess as well as to account for the correct relic density of dark matter. Note that the benchmark points predict a direct detection cross section which is not far from the actual experimental reach, and will likely be accessible in future experiments. \begin{table} \renewcommand{\arraystretch}{1.1} \centering \begin{tabular}{|ccccccc|} \hline Benchmark & $\sigma_{\gamma Z}$ & $\sigma_{\mathrm{MET}+j} $ & $ \sigma_{\gamma \gamma}$ &$\sigma_{jj}$ &$ \langle \sigma v \rangle_{\gamma \gamma} $ & $\sigma_{SI}$ \\ \hline & $< 3.5$\,fb & $< 6 $\,fb & $< 2 $\,fb& $< 10^3 $\,fb & $< 10^{-28} \frac{\mathrm{cm}^3}{\mathrm{s}}$ & $ < 4 \times 10^{-45} \mathrm{cm}^2$ \\ \hline $p_1$ & 0.86 & 3.7 & 1.4 & 1.3 & $ 3.9 \cdot 10^{-32} $& $6.9 \cdot 10^{-46}$ \\ $p_2$ &\alb{3.6} & 3.5 & \alb{6.0} & 1.4 & $5.5 \cdot 10^{-32}$ & $4.6 \cdot 10^{-46}$ \\ $p_3$ & 0.3 & 1.2 & 0.48 & 0.14 & $4.1 \cdot 10^{-32} $& $2.3 \cdot 10^{-46}$ \\ $p_4$ & 1.1 & 1.2 & 1.8 & 0.13 & $ 6.2 \cdot 10^{-32} $& $ 1.6 \cdot 10^{-46}$ \\ \hline \end{tabular} ]]>

20 \gev, \eta_j < 2.5$, while for the $\sigma_{\mathrm{MET}+j}$ we impose a cut of $ p_T^j > 500 \gev$. ]]>

500$\,GeV bin, which gave the most stringent constraints at 8\,TeV\@. A back of the envelope estimate indicates that the benchmark point $p_1$ (with large MET+$j$ cross section) should be within reach with a few fb$^{-1}$ at 13\,TeV\@. Benchmark points $p_3$ and $p_4$ (with small MET+$j$ cross section) would instead need few tens of fb$^{-1}$ to be excluded. We then argue that essentially all the viable portion of parameter space in figure~\ref{xsecplot2} should be within the reach of LHC-13TeV with $\lesssim100$\,fb$^{-1}$ of luminosity. A more detailed analysis, which we leave for a future work, is necessary in order to extract more precise values for luminosity needed to explore the allowed parameter space in our model. The final state $Z\gamma$ is also a promising channel. However, note that by tuning the couplings $g_{BB}$ and $g_{WW}$ one can generically suppress this branching ratio,\footnote{It scales like $\sim (g_{BB}-g_{WW})^2$.} and thus the signal. Hence the $Z\gamma$ is not a generic prediction of our model, in contrast to MET+$j$. Interestingly, dark matter experiments are also going to be able to probe our model. The future direct detection experiments should reach a sensitivity of approximately \linebreak $10^{-46}$\,cm$^2$ for spin independent cross section assuming a dark matter mass of around $300$\,GeV (see e.g.\ XENON1T prospects~\cite{Cushman-ml-2013zza}), which is in the ballpark of the predictions for our benchmark points (see table~\ref{tab:constraints}). In fact, future direct detection experiments should be able to probe most of the parameter space of the model which features a large width and is compatible with LHC-8TeV MET+$j$ searches (i.e.\ the region illustrated in figure~\ref{xsecplot2} with $g_{GG} \lesssim 0.3$), as the direct detection cross section is set essentially by $g_{GG}$ and $g_{\rm DM}$ (see eq.~\eqref{DDsigma}). Note that both MET+$j$ and direct detection DM cross sections can be reduced by decreasing the value of the coupling $g_{GG}$. However, in order to maintain a significant yield in the $\gamma \gamma$ channel, this should be accompanied by an increase of $g_{BB}$, pushing the model into a somehow less appealing region of the parameter space (especially from the point of view of the UV completion). We conclude that the simplified dark matter model we presented here provides sharp phenomenological predictions that can be further scrutinized in both LHC-13TeV and in future searches for galactic dark matter. We leave a more complete exploration of the parameter space of this scenario and the possibility of embedding it into UV complete models beyond the Standard Model for future investigations. ]]>