Scalar Dark Matter: Direct vs. Indirect Detection

We revisit the simplest model for dark matter. In this context the dark matter candidate is a real scalar field which interacts with the Standard Model particles through the Higgs portal. We discuss the relic density constraints as well as the predictions for direct and indirect detection. The final state radiation processes are investigated in order to understand the visibility of the gamma lines from dark matter annihilation. We find two regions where one could observe the gamma lines at gamma-ray telescopes. We point out that the region where the dark matter mass is between 92 and 300 GeV can be tested in the near future at direct and indirect detection experiments.


I. INTRODUCTION
The possibility to describe the properties of the dark matter (DM) in the Universe with a particle candidate is very appealing. This idea has motivated the theory community to propose a vast number of dark matter candidates and today we have many experiments searching for these candidates. The traditional way to look for dark matter is through direct detection where one expects to see the recoil energy from the scattering between the dark matter candidate and nucleons [1], or from the scattering between the electrons and the dark matter. One also could see exotic signatures at the Large Hadron Collider (LHC) associated with missing energy due to the production of dark matter. However, since one cannot probe the dark matter lifetime at colliders, this latter possibility is perhaps not the most appealing one. See Refs. [2][3][4][5] for reviews on dark matter candidates and corresponding experimental searches.
The annihilation of the dark matter in the galaxy into gamma rays can provide a very striking signal which can be used to determine the dark matter mass and understand the dark matter distribution in the galaxy. One expects more photons in the center of the galaxy and the dark matter profile dictates how many photons one could expect in other regions of the galaxy for a given value of the annihilation cross section. Since the dark matter candidate does not have electric charge, the dark matter annihilation into monochromatic photons occurs at loop level, and it could be very difficult to observe these lines due to the continuous spectrum. See Ref. [6] for a recent review on dark matter annihilation into gamma rays.
In the simplest dark matter model one has only a real scalar field [7], which is stable due to the existence of a discrete symmetry. This model has only two parameters (relevant for the DM phenomenology) and one can have clear predictions for direct and indirect detection experiments.
Since this is the minimal theory for dark matter one should investigate all the predictions to understand how to test this model in the near future. This model has been investigated by many groups [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. However, only recently it has been pointed out [24] that one can observe the gamma lines from dark matter annihilation in this context due to the fact that the final state radiation (FSR) processes are suppressed in some regions of the parameter space. This is by far not generic given a dark matter model, as photons from tree-level processes tend to dominate the spectrum.
In this article we revisit the singlet dark matter model investigating all current constraints from relic density, invisible Higgs decays, direct and indirect detection. Here we complete the study presented in Ref. [24]. Our main aim in this article is to present the scenarios that can be tested using direct and indirect detection experiments. We focus on the discussion of gamma-ray lines and point out that two regions in agreement with all current experimental constraints exist where the final state radiation processes are suppressed with respect to the annihilation into photons. Thus, there is hope that actually a line could be seen over the continuum background in these regions and more information about the dark matter particle could be extracted. We do not discuss the reach of proposed future experimental searches in detail but rather focus on general features necessary to be able to distinguish lines from the continuum.
This article is organized as follows. In Section II we discuss the main properties of the simplest spinless dark matter model as well as all relevant experimental constraints including the relic density.
In Section III we discuss in great detail the possible gamma lines in this model and the correlation between the gamma rays coming from final state radiation and the annihilation into γγ and Zγ.
In Section IV we summarize our main results. In the appendix we list all relevant formulas used in this article.

A. The Model
In the scalar singlet dark matter model (SDM) the dark matter candidate is a real singlet scalar field S which interacts with the Standard Model (SM) particles through the Higgs portal [7]. The Lagrangian of this model is very simple and is given by where H ∼ (1, 2, 1/2) is the SM Higgs boson and L SM is the usual SM Lagrangian. Once the SM Higgs acquires a vacuum expectation value, H = v 0 / √ 2 where v 0 = 246.2 GeV, the physical mass of the dark matter candidate reads as As is usually done, we assume a discrete Z 2 symmetry to guarantee dark matter stability. Under this symmetry S → −S such that all odd terms in the scalar potential are forbidden.
Once the electroweak symmetry is broken, the dark matter candidate S can annihilate into all Standard Model particles through the portal coupling λ p . In this model one has only two relevant parameters for the dark matter study, the physical dark matter mass M S and the Higgs portal coupling λ p . This is the reason why one can make definite predictions in this model once the relic density constraints are used. This model can be considered as a toy model for dark matter, but also is the perfect scenario to understand the possible predictions for different experiments and their interplay.

Higgs Decays
The most conservative, model-independent limit on the Higgs invisible decay branching ratio is set by CMS to be BR(h → inv) < 0.58 [26]. However, if we study the predictions for the invisible Higgs decay in a particular model, the situation can be rather different. In the scalar singlet dark matter model, there is no modification to Higgs physics at the LHC apart from a possibly large invisible decay to dark matter if allowed kinematically. Since also the Higgs production cross section is unaffected in this model, the invisible width modifies the signal strength of the Higgs decay to a P 1 P 2 final state in the following way: The combined limit from the final states W W * , ZZ * , γγ,bb, τ + τ − is given by R total = 1.17 ± 0.17 [27]. This leads to a 95% confidence upper bound on the invisible Higgs branching ratio of This bound is valid for any model which only modifies the Higgs invisible branching ratio. Note that a statistically significant deviation of the combined signal strength above one, R total > 1, would rule out this simple dark matter model up to M S = M h /2. The bound obtained here is used when we later show the allowed parameter space in the low-mass region, see Figs. 6 and 7 for details.

Relic Density
In order to compute the relic density of our dark matter candidate S, we use the analytic approximation [28] where M Pl = 1.22 × 10 19 GeV is the Planck scale, g * is the total number of effective relativistic degrees of freedom at the time of freeze out, and the function J(x f ) reads as The freeze-out parameter x f = M S /T f can be computed by solving where g is the number of degrees of freedom of the dark matter particle. Details on the calculation of the cross sections for the different DM annihilation channels and the corresponding analytic formulas, including the expressions to perform the thermal average of the cross section times relative velocity σv rel , can be found in Appendix B.
In Fig. 1   the corresponding coupling λ p that results in today's full DM relic density, Ω DM h 2 = 0.1199 ± 0.0022 [29]. As expected, in the low mass regime the dominant channels are bb and τ + τ − , and after threshold the annihilation into W + W − and ZZ become dominant. Below M S = 150 GeV, we calculated the cross sections from the tabulated partial Higgs widths [30], such that three-and four-body decays of the gauge bosons below threshold as well as QCD corrections are included; see Appendix B for details. In the high-mass regime, the contributions from the annihilation into the SM Higgs h and top quark pairs are significant.

Direct Detection
To discuss the possible constraints from dark matter direct detection experiments we need to know the elastic nucleon-DM cross section. In the scalar singlet DM model, the spin-independent nucleon-DM cross section is given by Fermi-LAT excluded 10 −50  where m N = (m p + m n )/2 = 938.95 MeV is the nucleon mass for direct detection, f N = 0.30 ± 0.03 is the matrix element [20], and µ = m N M S /(m N + M S ) is the reduced nucleon mass.
In Fig. 3 we show the predictions for the spin-independent nucleon-DM cross section σ SI for the typical choice f N = 0.30 and the corresponding experimental bounds. This model is very simple and one can predict clearly the values for the elastic cross section once the relic density constraints are imposed. As it is well known, the experimental bounds assume that the dark matter particle under study makes up 100% of the DM of the Universe. An important observation is that around the Higgs mass resonance direct detection experiments are not able to probe the parameter space in the near future as apparent in Fig. 3. However, as will be shown later, indirect searches are particularly sensitive to the resonant region and thus highly complementary to direct detection experiments. The projected limits by XENON1T [32] tell us that one can test this model for a dark matter mass up to a few TeV.
Note that the current limit from LUX [31] on the scalar singlet DM model strongly depends on the particular value that is chosen for f N . In Fig. 4, we show the predictions for

Missing Energy Searches
As it is well known, one can hope to observe missing energy signatures at colliders from the presence of a dark matter candidate. We have discussed the low mass region where using the invisible decay of the Standard Model Higgs one can constrain a small part of the allowed parameter space in this model. Unfortunately, in the resonance region the invisible branching ratio of the Higgs can be very small and one cannot test this model in the near future. In the heavy mass region one can use mono-jets and missing energy searches where one produces the scalar singlet through the Standard Model Higgs. Unfortunately, in this case the production cross sections are small and this analysis is very challenging. See Ref. [35] for a recent discussion. 100 1000   0.1199 ± 0.0022 [29]. In green we show the bounds from the invisible decay of the SM Higgs, using the CMS bound BR(h → SS) < 58% [26], as well as the calculated limit from Eq. (4). The red part of the relic density curve is excluded by the LUX direct detection experiment [31], while the blue part of the curve shows the projected reach of the XENON1T experiment [32]. The orange part of the curve is excluded by the bb limits from Fermi-LAT [34].

Summary
In Fig. 6 we show the allowed parameter space in agreement with the relic density constraints,  Fig. 6 we also show the experimental bounds on the invisible decay of the SM Higgs [26] and the projected direct detection bounds from the XENON1T experiment [32]. The gray region is ruled out by the relic density constraints because in this region one overcloses the Universe having too much dark matter relic density. Notice that even this simple model for dark matter is not very constrained by the experiments.

Low Mass Regime
In the low mass region the allowed dark matter mass is 53 GeV ≤ M S ≤ 62.8 GeV. In this region close to the Higgs resonance the dark matter can annihilate into Standard Model fermions or into two fermions and a gauge boson. In Fig. 7 we show a detailed analysis of this region to understand which part of the parameter space is ruled out by experiments. Notice that the main annihilation channel is SS →bb. In this model one can set bounds only using the constraints on the nucleon-DM cross section. The scattering between electrons and DM is highly suppressed by the small Yukawa coupling.
From the results presented in Fig. 7 one can see that the resonance region cannot be excluded or tested in the near future by direct detection experiments. This is a pessimistic result, but fortunately this region can be tested at gamma-ray telescopes as we will discuss in the next section.

Heavy Mass Regime
When the dark matter is heavy it can annihilate into all Standard Model particles. For

A. Final State Radiation
There are three relevant regions which define the properties of the gamma spectrum coming from dark matter annihilation. Let us define x γ = E γ /M S , where E γ is the energy of the photon and M S is the dark matter mass. When x γ is very small one has the photons coming mainly from hadronization, i.e., the dark matter annihilates into quarks and from the cascade one has the photons with a continuous spectrum. When x γ is close to one, one finds that the final state radiation processes contribute more because they can provide hard photons. Finally, when x γ = 1 one has the gamma line with energy equal to the DM mass. Therefore, one must understand the final state radiation processes to investigate the visibility of the gamma lines.
The relevant final state radiation process for our study is SS →XXγ, with the kinematic endpoint of the continuous γ spectrum at in the non-relativistic limit s = 4M 2 S . In the low-mass regime the dominant process is SS →f f γ with the strongest contribution from the bottom quark, while in the high-mass regime SS → W + W − γ becomes dominant. The differential cross section times velocity of those processes is given by where the integration limits for the integration over E 1 for a fixed E γ are given by in the limit s = 4M 2 S . See Appendix C for the amplitudes of the two relevant processes for final state radiation, SS →f f γ and SS → W + W − γ. Notice that in the low-mass region the FSR is suppressed by small Yukawa couplings. Therefore, this is the region where generically one can have a visible gamma line.

B. Gamma Flux
The differential photon flux is given by where the factor J ann contains the astrophysical assumptions about the DM distribution in the galaxy and thus all the astrophysical uncertainties. Here n γ is the number of photons per annihilation, and dN γ /dE γ is the differential energy spectrum of the photons coming from dark matter annihilation. In all numerical calculations, we will use the J-factor from the R3 region-of-interest, given by the Fermi-LAT collaboration to be J ann = 13.9 × 10 22 GeV 2 cm −5 [38]. The R3 region is a circular region of radius 3 • centered on the galactic center [38]. The differential flux of the line is extremely narrow, however to make connection with the experiment it will be folded with a Gaussian function modeling the detector resolution.
• SS → γγ: for the annihilation into two photons the flux is given by with The parameter ξ is a measure of the detector energy resolution which varies between 0.01 and 0.1 in the relevant energy range. The factor w = 2 √ 2 log 2 ≈ 2.35 determines the full width at half maximum as σ 0 w = ξE 0 , therefore we have σ 0 = E 0 ξ/w in the usual Gaussian function. For the annihilation to γγ, the energy of the gamma line is at the dark matter mass, • SS → Xγ: for the annihilation into an unstable final state particle along with a photon, the flux is given by Here Γ X is the decay width of the unstable particle in the final state and M X is its mass.
See Appendix D for a derivation of the differential energy spectrum used in Eq. (17). The gamma line energy is given by Using these expressions for the differential flux we now study the predictions for the gamma lines in this model in the benchmark scenarios defined in Table I.
In Fig. 8 (a) we show the gamma spectrum for the scenario 1 with M S = 62.5 GeV. In this case one has the resonant dark matter annihilation through the SM Higgs. As one can see in this scenario it is possible to identify the gamma line from DM annihilation into γγ, while the line from Zγ is not visible in the plot since it is at x γ = 0.47 and will be swamped in the FSR background.
The main contribution to final state radiation in this case is coming from the annihilation intobbγ but it is suppressed by the small bottom Yukawa coupling. Therefore, in this case one has a large difference between the final state radiation and the gamma line. (a)   In Fig. 8   is a large difference between FSR and the gamma lines because the endpoint of the FSR is far from the DM mass.
The case when the DM mass is 316 GeV is shown in Fig. 8 (c). There is a large difference between the rate for the Zγ and γγ lines. Unfortunately, in this case one could see the lines only with a perfect energy resolution. The cross section for the final state radiation processes is large in this case making the observation of gamma lines very challenging. Finally, we present in Fig. 8 (d) the energy spectrum for the case when M S = 500 GeV. In this case one cannot distinguish the gamma lines since the difference between the final state radiation and the gamma line is very small.
In order to have a more generic discussion about the visibility of the gamma lines in Fig. 9 we show the ratios between the γγ and Zγ fluxes and the gamma flux from final state radiation. We display these ratios for M S ≥ 100 GeV. We show the curves for 1% energy resolution (green solid), 5% energy resolution (red dashed), and 10% energy resolution (blue dotted). To be conservative we can say that the lines are visible if the ratio between the fluxes in Fig. 9 is larger than a factor 10. This means that for realistic experiments with an energy resolution of 5 %, the gamma lines can be visible when the dark matter mass is smaller than 300 GeV.
As one can appreciate from the above discussion one could observe the gamma lines from dark matter annihilation in this model only when the dark matter mass is small, i.e., In the scalar singlet DM model the Standard Model Higgs can decay into dark matter in the low mass region. The invisible decay width of the Higgs in this case is given by To calculate the invisible branching fraction with where K 1 and K 2 are modified Bessel functions of the second kind.
For M S ≥ 150 GeV we use the tree-level expressions calculated below, since then loop corrections overestimate the tabulated width. See the discussion in Ref. [20] for more details.
From the cross section σ, the thermal average can be computed via • Annihilation into Standard Model fermions: where N f c is the color factor of the fermion f .