Scalar Dark Matter: Real vs Complex

We update the parameter spaces for both a real and complex scalar dark matter via the Higgs portal. In the light of constraints arising from the LUX 2016 data, the latest Higgs invisible decay and the gamma ray spectrum, the dark matter resonant mass region is further restricted to a narrow window between $54.9-62.3$ GeV in both cases, and its large mass region is excluded until $834$ GeV and $3473$ GeV for the real and complex scalar, respectively.

In the light of the direct detection arising from the LUX 2016 data, the indirect detection arising from the Higgs invisible decay at LHC [29], and the indirect detection arising from gamma ray spectrum induced by DM annihilation at Fermi-LAT [30,31] and HESS [32], the DM mass regions are updated as follows. (i) For the real scalar DM the resonant mass and large mass region is modified to 54.9 ≤ m s ≤ 62.3 GeV and m s ≥ 834 GeV, respectively.
(ii) For the complex scalar DM the resonant mass is similar to the real scalar DM and the large mass region is modified to m s ≥ 3473 GeV. See Table.II for details. Although we will not discuss N copies of real scalar with N ≥ 3, it can be inferred that the large mass region for these choices is excluded, and the resonant mass region will be further suppressed as N increases.
The plan of this paper is organized as follows. Sec. II is devoted to address the notation in the real and complex DM models. In Sec. III we discuss the direct detection at the LUX experiments, where the update will be noted. In Sec. IV we discuss the indirect detections at the LHC, Fermi-LAT and HESS. If available, we will compare the differences between the real and complex scalar DM. Finally, we conclude in Sec.V.

II. SCALAR DARK MATTER
The Lagrangian for the dark sector in the DM model is given by, where µ is the DM bare mass, and the last term denotes the interaction between DM sector and SM Higgs, with κ referring to the Yukawa coupling constant. In order to keep the DM stable and eliminate harmful operators, one simply imposes the following Z 2 parity on the full Lagrangian, Below the electroweak scale, Eq.
(1) is rewritten as, Here m 2 s = µ 2 + κυ 2 /2 is the square of DM mass, and H = (υ + h)/ Dark matter s can be composed of a single or multiple real scalar components. In this paper, we focus on the following two models, which corresponds to a real and complex scalar, respectively. Note that the masses for s 1,2 in the model B are degenerate. In some situation beyond Eq.
(1), it may include a small mass mixing term, which directly leads to non-degenerate masses. Mixing effects will be neglected in the following discussion.

III. DIRECT DETECTION
Let us firstly discuss the direct detection on the model A and B in the light of the latest LUX data. Previous discussions can be found in Ref. [38] and Ref. [39] for the model A and where m N is the nucleon mass,μ = m N m s /(m N + m s ) is the DM-nucleon reduced mass, and f N is the hadron matrix element. Note that the factor c(A) = 1 and c(B) = 2 for the model A and B, respectively.
In fig.1 we show the plots of σ SI as the function of DM mass by virtue of the code MicrOMEGAs [37], where the dependence of σ SI on Yukawa coupling constant κ in Eq. (5) is eliminated by the constraint from DM relic abundance. Input values for parameters in Eq.(5) are shown in Table I [38,39]. Consequently, the ability of exclusion is expected to be stronger in our discussions.
For the model A, we find that the large mass region is obviously uplifted to 834 GeV (LUX 2016) from 185 GeV (LUX 2015). In contrast, there is only about ∼ 1 GeV deviation Ω DM h 2 = 0.1199 ± 0.0027 [33] f N 0.3 [28] m h = 125 GeV [34,35] Table I, the Yukawa coupling κ should be multiplied by √ 2, as inferred from that Ω DM h 2 is proportional to 1/κ 2 . We have verified this by the numerical calculations in terms of MicrOMEGAs.
Second, the factor c(B) is two times of c(A). Therefore, the DM-nucleon scattering cross section in the model B is roughly ∼ 4 times of that in the model A in the large mass region.
However, in the resonant mass region where σ SI is small, the enhancement effect is not so obvious as in the large mass region.
Other direct detections on either the resonant mass region or the large mass region in fig.1 are rather insufficient. The discovery of collider signatures requires extremely large integrated luminosity at least of order O(10) ab −1 at the 14-TeV LHC [40]. The discovery of astrophysical signatures requires the DM scattering cross section relative to the DM mass, σ/M at least of order 10 −7 cm 2 /g [41], in contrast to present limits typically of order 0.1cm 2 /g. Some cosmological considerations may impose interesting constraints on these models. See, e.g., Ref. [42].

IV. INDIRECT DETECTION
In this section we discuss indirect detections arising from Higgs invisible decay at the LHC as well as the gamma ray spectrum at the Fermi-LAT and HESS. In contrast to the large mass region, these experiments may impose strong constraints on the resonant mass region.

A. Higgs invisible decay
When the DM mass is smaller than m h /2, the measured Higgs invisible decay at the LHC imposes strong constraint on the decay width Γ(h → ss). For details on the calculation of Γ(h → ss), see, e.g., [42]. The latest LHC result has been updated as [29], where the SM Higgs decay width Γ h 4.15 MeV. In fig.2 we show this constraint on the DM mass, which indicates that for the model A the DM mass lower bound in the resonant region has been modified to 52.3 GeV from 51.8 GeV [40], and for the model B the value of DM mass lower bound in the same region is about 51.5 GeV.

B. Gamma ray
Now we discuss the indirect detection arising from gamma ray spectrum at Fermi-LAT and HESS. These experiments impose upper bounds on the magnitudes of thermal averaged DM annihilation cross sections times velocity υ rel . Consider that the maximal value of these cross sections in our models corresponds to the DM mass near m h /2, the resonant mass region is mostly sensitive to these experiment limits. In the following discussion we will use the gamma ray limits on < σ γγ υ rel >, for which earlier analysis in the model A can be found in [43][44][45][46].
The value of < σ γγ υ rel > is calculated via the standard formula [47], where x = m s /T , s is the square of the center-of-mass energy, and K1 and K2 are modified Bessel functions of the second kind. For the details on the calculation of σ γγ υ rel in Eq. (7), see the appendixes in Refs. [43,44]. Similar to the DM-nucleon scattering cross section the value for σ γγ υ rel in the model B will be doubled in comparison with A, which implies that the gamma ray limits will be more sensitive to this model.
In fig.3 we present the gamma ray constraint on the DM mass. The Fermi-LAT R3 and R41 limits are both shown simultaneously. For the model A, the Fermi-LAT R3 limit excludes DM mass above 62.3 GeV in the resonant mass region, while the HESS limit is not so sufficient as the LUX limit in the large mass region. This result is consistent with the earlier one obtained in [44].