Exploring the Potential of the Dark Matter Candidate from the Madala Hypothesis with Multi-frequency Indirect Detection
The Madala hypothesis was proposed by members of the Wits-ATLAS group to account for several anomalies in both ATLAS and CMS data at the LHC. This hypothesis extends the standard model through the addition of two scalar bosons and a hidden sector that can provide a dark matter candidate. This hidden sector interacts with the standard model only through the mediation of one of these scalars . The couplings of are not amenable to investigation in current collider data and so are assumed to be Higgs-like to reduce the parameter space of the model. Our previous work  has shown that these couplings can be limited via indirect dark matter detection experiments in gamma-rays (for resonant annihilations into ). Here we will treat the dark matter and masses independently, and we generalise our previous work  and examine what fraction of the cosmological dark matter abundance can be accounted for by particles in the hidden sector of the Madala hypothesis dark matter when these annihilate to standard model products via a Higgs-like . We will also extend our gamma-ray analysis of Madala hypothesis dark matter to include the constraints of diffuse radio data from the Coma galaxy cluster in addition to the Fermi-LAT gamma-ray data from both this target and the Reticulum II dwarf galaxy.
Our analysis indicates that either the Madala hypothesis cannot provide the bulk of cosmologically relevant dark matter, or the boson cannot be simply Higgs-like. These apply unless the candidate particle exceeds a mass of GeV. Both these scenarios may reduce the attractiveness of the hypothesis as the second case will imply that many free parameters must be added to describe , greatly weakening fit significances for the model. To investigate the full consequences of this further work will necessitate using larger astrophysical data sets to strongly constrain details about .
The Madala hypothesis has three important constituents: a heavy Higgs-like Madala boson , a mediator scalar , and a Dark Matter (DM) candidate . This hypothesis was put forward to explain anomalies seen in both ATLAS  and CMS , particularly in the transverse momentum of the Higgs boson as well as event excesses in multi-lepton final states [4, 5, 6, 7]. The scalar mediator is introduced to mitigate problems in quartic couplings , through it, can interact both with and the Standard Model (SM). Given that the run-1, preliminary run-2  and the latest run-2 releases , data from the Large Hadron Collider (LHC) did not remove the excesses attributed to the Madala particles, and that it provides a candidate for the missing content of the universe, it is worthwhile to examine the properties of the model from an astrophysical standpoint.
Here we can investigate the properties and thus those of , which cannot be strongly limited by current collider data , through methods of indirect DM detection. That is, we can place limits on the properties of pathways from to the SM by predicting resulting fluxes of gamma-rays, synchrotron and Inverse Compton (IC) emission within cosmic structures and comparing these to known spectra/upper-limits for these target environments.
In this work we will use diffuse radio data from the Coma galaxy cluster  as well as Fermi-LAT  gamma-ray limits on both Coma  and the Reticulum II dwarf galaxy  to examine the consequences of the simplifying assumptions used to describe . This being that has Higgs-like couplings to the SM. In order to model this we will use decay branching data for Higgs-like particles from . We will do this by determining confidence level upper-limits on the SM annihilation cross-section and comparing these to the canonical relic values . If the derived limits rule out the allowed relic values, then it is sufficient to say that cannot constitute all cosmologically relevant DM. We then take the ratio of the derived cross-section with lower-limit of the relic band to determine the maximal allowed fraction of the cosmological abundance of DM that can be composed of particles from the Madala hypothesis. Unlike our previous work , where we study whether astrophysical data allow for Higgs-like , we do not study only the resonant case , we allow and to be independent. We will also extend the work from  by determining upper-limits on the branching ratio of to bosons if is assumed to constitute all of DM by making use this generalised mass scenario and by including radio data. We choose the boson channel as it was the most promising in previous analysis .
We find that, with both Coma radio and Reticulum II gamma-ray data, we can limit particle DM with a Higgs-like mediator to (10%) of the cosmological DM abundance, and can rule out Higgs-like couplings to bosons for a broad range of masses, - GeV, using both radio and gamma-ray data.
2 Dark Matter Annihilation
The Madala DM particles annihilate to bosons that can decay to SM particles . The annihilation cross-section found by astrophysical probes will be an effective one, from SM.
The source function for particle (electrons/positrons or photons) with energy from a annihilation and subsequent decay is taken to be
where is distance from the halo centre, is the non-relativistic velocity-averaged annihilation cross-section, labels the annihilation channel intermediate state with a branching fraction and differential -particle yield , is the radial density profile of particles in the halo, and is the mass. The channels used will be quarks , electron-positron , muons , -leptons , bosons , bosons , and photons .
3 Indirect Detection in Coma and Reticulum II
For the DM-induced -ray production, the resulting flux calculation takes the form
with being the source function for frequency and position within the given DM halo at redshift , and is the luminosity distance to the halo. In the case of Reticulum II, we will instead calculate the resulting gamma-ray flux based on the astrophysical J-factor:
with being the halo density profile, the integral being extended over the line of sight , and is the observed solid angle. The flux is then found to be
The halo of Reticulum II is found to have GeV cm .
For the Coma galaxy cluster, the local emissivity for the emission mechanism (synchrotron, ICS, bremsstrahlung) can then be found as a function of the electron and positron equilibrium distributions as well as the associated power (for power functions see [22, 23])
In Coma we will assume the following halo data following : the virial mass is given by M, with virial concentration , and that the density profile is of the Navarro-Frenk-White (NFW) form . The thermal electron distribution and magnetic field profiles are taken from the best-fit values of  and  respectively. We also use an annihilation flux boosting factor from dense halo substructure in Coma that is following methods described in [28, 29]. We will show results both with and without this boosting factor.
Taking a spectral function , we can compare it to data from the Coma cluster and Reticulum II and find confidence level exclusion limits on the value of for the process SM.
4 Results and Discussion
Here we display the fraction of the cosmological DM abundance accounted for by Madala model DM. This is derived by determining the annihilation cross-section limits placed on the decay of Higgs-like coupled , for a range of masses, by the spectra of the Coma galaxy cluster (radio and gamma [9, 11]) and the Reticulum II dwarf galaxy gamma-ray spectrum . If this cross-section falls below the range of the canonical thermal relic value ( - cm s)  then the ratio of the derived limit and lower end of the canonical band is taken to be the maximal fraction of DM accounted for. As the candidate model is already limited to annihilating too slowly to constitute all of the observed present epoch abundance.
In the left-hand panel of Figure 1 we show the DM fraction limits imposed on by the Fermi-LAT gamma-ray data. These indicate that Coma can only constrain this fraction for GeV when the substructure boosting factor is used. However, the Reticulum II spectrum places strong limits that force to provide % of the DM abundance while GeV. These limits become weaker as the mass increases because the upper-limits in Reticulum II rise as a power-law for higher frequencies and the peak of DM-induced spectrum shifts with .
In the right-hand panel of Fig. 1 we see that the radio limits are significant provided a boost factor of from substructure is assumed. Since this is conservative within the literature, and based on robust halo simulations , it is not an undue assumption in a structure as large as the Coma cluster. The limits in this case are very similar to those from Reticulum II. However, they become stronger for larger , due to fact that the harder resulting spectra conflict with the spectral profile of diffuse radio emissions in Coma (due to increased boson production at larger ).
For a very general analysis of the Madala hypothesis DM candidate, under the same simplifying assumptions used to limit the space of free-parameters in the model  (that is Higgs-like), we have shown that current data from the Reticulum II dwarf galaxy and the Coma galaxy cluster is sufficient to limit the possible abundance of Madala-associated DM particles to (10%) if GeV. This expands on previous work  showing that it may be possible to use gamma-ray data to place limits on the couplings of to the standard model, and possibly even rule-out a Higgs-like if Madala DM is to constitute the entire cosmological abundance. Here we have also removed some of less general assumptions from the prior work (that only resonant production was considered).
Using the same modelling techniques as applied here, we will be able to generalise the analysis of  to also allow the and masses to be independent, opening a much broader capacity to constrain the couplings of to the SM. This is in addition to the inclusion of radio data, which is shown to have similar independent constraining power to that of the previously employed gamma-ray data. We demonstrate this generalisation below in Figure 2. This displays the confidence level limits on the branching ratio of to bosons from both gamma-rays (left panel) and radio data (right panel) when thermal relic annihilation cross-sections are assumed . This is displayed as a ratio of the branching ratio to that of the required fraction for a Higgs-like particle of mass . We see that the conclusion of  is not substantially weakened by this generalisation, with evidence limiting the possibility of a Higgs-like boson coming from both Reticulum II gamma-ray data and Coma diffuse radio data while GeV.
By bringing the viability of the simplifying Higgs-like assumption into question we can see that either many more free parameters must be added to the Madala model, which will require further astrophysical data to constrain accurately and may lower the significance of existing fits to excesses, or the particle of the Madala hypothesis must take a back seat as a candidate to explain all of DM. Both of these scenarios may well serve to weaken the attractiveness of the proposed Madala model. However, a final possibility is that the hidden sector particle is of large mass, GeV, limiting its ability to be put forward as an explanation in scenarios like the galactic centre gamma-ray excess  as well as its detectability in collider experiments. This work remains especially relevant due to the persistence of the motivating LHC excesses into run-2 data .
This work is based on the research supported by the South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation of South Africa (Grant No 77948). G.B. acknowledges support from a post-doctoral grant through the same initiative and institutions. We acknowledge useful discussions with B. Mellado on the Madala hypothesis proposed by members of the Wits-ATLAS group to account for several anomalies in both ATLAS and CMS data at the LHC.
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