The galactic center excess from ℤ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb{Z}}_3 $$\end{document} scalar semi-annihilations

We investigate the possibility of the ℤ3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb{Z}}_3 $$\end{document} scalar singlet model explaining the Fermi galactic centre excess. We find a good fit to the measured spectral excess in the region where the dark matter mass is comparable to the Higgs and the Higgs portal coupling λHS ∼ 0.04. This preferred region is consistent with constraints from vacuum stability and current dark matter experiments, and will be discovered or falsified soon by future dark matter direct detection experiment.


Introduction
The nature of dark matter (DM) is one of the biggest questions in contemporary particle physics. Despite intense efforts at many different experiments, no unambiguous nongravitational signal has been found. However, current and near future experiments offer the prospect of decisively testing the weakly-interacting massive particle (WIMP) hypothesis. This makes experimental excesses more theoretically attractive now than in the past.
One of the most interesting recent anomalies in dark matter searches is the γ-ray galactic centre excess (GCE) discovered in Fermi data after subtraction of backgrounds. The original discovery [1] has been corroborated by several theoretical analyses [2][3][4][5][6][7], and recently by the Fermi collaboration itself [8]. The GCE has many of the features expected of DM annihilation into Standard Model (SM) states: the morphology in the sky matches what is expected from DM density distributions, and the required cross sections are very close to that of the canonical thermal WIMP. The spectrum is easily fit by various SM final states; bb for DM masses of 30-60 GeV offers the best fit, but Higgs, gauge boson and top final states with larger DM masses are also acceptable [7,9,10] . This has inspired considerable model building efforts to explain the observed signal . Alternative explanations have been advanced, including pulsars [32][33][34][35][36][37][38] or cosmic rays at the galactic centre [39][40][41], but the efficacy of these explanations is contested [5,[42][43][44]. It is worth noting that the studies of refs. [37,38] find evidence for a point source origin of the GCE. In any case, we find it interesting to consider a DM interpretation of this signal.
Semi-annihilation (SA) is a generic feature in DM phenomenology that occurs whenever DM is stabilized by a symmetry larger than Z 2 .It specifically modifies the relic density and indirect detection signals, which makes it interesting for interpreting the GCE. We compare SA and ordinary DM annihilation in figure 1. SA is characterised by non-decay processes with an odd number of external dark-sector states. In addition to enabling different final states and kinematics, SA is also irrelevant for collider and dark matter searches. This can allow strong indirect signals while weakening other constraints. In a previous work [45], we showed that SA in a two-component DM model could explain the GCE through processes where a single Higgs is produced nearly at rest. 1 The subsequent decay of the Higgs to bb produces a spectrum very similar to annihilation of 60 GeV DM to the same final state. The dark sector only coupled to the SM through the Higgs portal, and SA played two essential roles related to this interatction. With SA significantly contributing to setting the thermal relic density, smaller Higgs portal couplings were allowed, thereby alleviating the stringent constraints from LUX. Additionally, DM annihilating through the Higgs portal will always preferentially produce gauge bosons over Higgses, resulting in a poorer fit to the GCE γ-ray spectrum. A large SA cross section substantially enhances the production rate of Higgses in our model.
In this paper, we apply the same approach to the simplest SA model: the Z 3 scalar singlet model (Z3SSM) [49]. This model extends the SM by a single DM particle only, coupled renormalisably to the visible sector through the Higgs portal. It is also a minimal extension of the well-studied Z 2 scalar singlet model [50]. The benefits that SA brought in our previous model apply also to the Z3SSM: weakening the direct detection bounds and enhancing the production rate of Higgses. The Z3SSM is a simpler and more constrained model and it is of interest to see whether it also enjoys the privilege to explain the GCE.
The outline of this paper is as follows. We begin by reviewing the Z3SSM and vacuum stability constraints in section 2. Our fit to the GCE is described in section 3, and other relevant constraints are given in section 4. Finally we give our conclusions in section 5.

Model
The Z3SSM is the simplest example of semi-annihilating dark matter. The SM is extended by one new particle, a gauge singlet complex scalar S. Its stability is ensured by a global Z 3 symmetry under which it is charged while all SM fields are neutral. This can be the low-energy remnant of a spontaneously broken U(1) gauge symmetry, provided that S is the only light degree of freedom [51]. The Lagrangian is

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There are four new parameters: the DM mass M S , SA coupling µ 3 , Higgs portal coupling λ HS and quartic λ S . Of these, µ 3 may be taken real and positive without loss of generality while the quartic couplings are perturbative if |λ HS | < 4π and |λ S | < π [52].
The vacuum stability of this model was studied in ref. [49]. There are four inequivalent vacua, depending on which of H and φ acquire VEVs. The desired solution has φ = 0 to preserve the global Z 3 , and H = 0 for EWSB. Demanding that this is the global minimum at the weak scale sets the bound The second inequality results from imposing the perturbativity constraint. This is the most robust consistency bound on the SA coupling. 2 Stronger bounds exist when we consider the effects of the renormalisation group. For the values of the Higgs-portal couplings of interest, λ HS ≈ 0.05, large values of λ S destabilize the vacuum as we run the couplings to a high scale Λ. Requiring that Λ 10 9 GeV, i.e. the same scale as the instability in the SM, gives the stronger constraint Aside from these vacuum stability constraints, the DM quartic λ S has no further effect on the phenomenology, leaving a 3-dimensional parameter space.

Galactic center excess
The Planck satellite measured the dark matter density Ω DM h 2 ∈ [0.1126, 0.1246] at 3σ [53]. We use micrOMEGAs 4.0 [54] to compute the relic density including the effect of SA, and demand that S saturate this result. This fixes µ 3 as a function of (M S , λ HS ). In this model, SA decreases the relic abundance, so there is an effective upper bound on λ HS as a function of M S given when the correct relic abundance is obtained for µ 3 = 0. This is the upper grey region in figure 3, for λ HS 0.06. We also show a lower grey region where we need µ 3 > 4000 GeV to produce the Planck results, and a gold line showing the consistency bound of eq. (2.3).
The differential photon flux from DM annihilation and SA is where ρ is the DM density and the sum i runs over different annihilation and SA channels. The astrophysical dependence of the flux factors out, and is conventionally expressed in terms ofJ We follow ref. [7] and parameteriseJ asJ

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whereJ can = 1.58 × 10 24 GeV 2 /cm 5 . This value corresponds to a Navarro-Frenk-White (NFW) [55] DM halo profile with scale radius r s = 20 kpc, slope γ = 1.2, and local dark matter density ρ = 0.4 GeV/cm 3 at r = r = 8.5 kpc. The integral (3.2) is evaluated over the Fermi region of interest, a 15 • × 15 • square centered on the galactic center. J ∈ [0.14, 4.0] represents our uncertainty in the DM density distribution, derived from the ranges γ = 1.2 ± 0.1 and ρ = 0.4 ± 0.2 GeV/cm 3 . We will find that larger values of J , and hence γ and ρ , are preferred. We note that we could also consider smaller values for the NFW parameters, together with a modest O(10) boost from DM substructure [56,57]. The question of whether, and how large, such an enhancement can be is an active area of current research [58][59][60][61][62]. However, we need a much smaller increase than the O(10 2 -10 3 ) boosts usually considered [61], which is more robust to arguments against the existence of such enhancements.
The sum in eq. (3.1) runs over all annihilation processes S † S → SM , plus the SA process SS → S † h. We use the PPPC 4 DM ID [63] expressions f i (m DM , E γ ) for the energy distributions. For the SA channel, we use the PPPC results for the hh final state, multiplied by one-half as only a single Higgs is produced. Additionally the Higgs is produced with an energy so we use the PPPC distributions for m DM = E h . We may then write the total photon flux as The Fermi-LAT collaboration presented a total of eight differential spectra of the residual photon flux after subtracting the interstellar emission and point-source contributions [8]. These correspond to four different models of (background) interstellar emission and two different models for the spectrum of the excess. The interstellar emission models in turn consist of two choices for the spatial distribution of sources within the galaxy, and two different fitting algorithms. The spatial distribution of sources was taken either to follow that of OB-stars (relatively close to the galactic centre) or pulsars (relatively far from the galactic centre), with the expectation that the actual distribution lies between the two. When fitting to the data, the Fermi collaboration considered intensity-scaled and index-scaled models for the background; the former varied the normalisation only, while the latter also varied the spectral index.
The two different choices for the spectra of the excess deserve some discussion. The simpler choice models the excess as a simple power-law with an exponential cut-off as a proxy for either pulsar emission or DM annihilation. The Fermi collaboration also considered a more complex fit where the energy band between 1 and 100 GeV was divided into ten bins of equal logarithmic size. The spectrum was then fit with a power-law where the overall normalisation and the spectral index within each bin were allowed to vary, such that the spectrum is continuous at all points.
We perform a χ 2 analysis on the eight computed photon spectra in 20 bins, assuming a symmetric uncertainty distribution. The quality of the fit improves with increasing J , especially once all constraints are considered. For spectra modelled with a single powerlaw throughout all energy bins, the best fits are found in the intensity-scaled Pulsars and OB-stars scenarios. The other two, the Pulsars (OB stars) index-scaled scenarios, are too hard for the Z3SSM. The best fit point for the pulsar (OB stars) intensity-scaled scenario with J = 4.0 is m S = 125(161) GeV, λ HS = 0.0467(0.0548), µ 3 = 52.7(64.5) GeV and χ 2 = 2.4(2.1). We show the flux at these points together with the GCE spectra in figure 2, and the 1, 2 and 3σ contours in figure 3.
As for the spectra modelled with a different power-law in each energy bin, only one can be reasonably fit with the Z3SSM: the index-scaled Pulsar scenario. The remaining three are too soft. The fit here is worse, but the best fit point has χ 2 = 16. Except the ones shown in figure 2 and figure 3, the fits with the other five spectra presented by the Fermi collaboration all have χ 2 /d.o.f. 4. The best fit points are characterised by a strong annihilation signal and a relatively small SA cross section. The contribution of the different final states W W : ZZ : hh : S † h ≈ 69% : 29% : 1.2% : 0.43%(45% : 20% : 27% : 8.0%) for Pulsars (OB stars). This arises as the SA cross section is bound by the perturbativity constraints on µ 3 . However, SA still plays an important role in setting the relic density for smaller λ HS , and thus avoiding direct detection constraints.

JHEP06(2016)156 4 Additional constraints
An unavoidable constraint for any potential explanation of the GCE comes from Fermi observations of dwarf spheroidal galaxies (dSphs) [64,65]. These are DM-dominated objects offering low backgrounds and reasonably well-understood density distributions. The same experiment that observed the GCE has seen no corresponding excess from these sources.
The most recent limits from Fermi [66] are given in terms of cross sections to various SM final states. Our model (semi-)annihilates to several different final states, so to apply these limits we make two simplifications. We assume that the differences between the spectra for W W , ZZ and h(h) are sufficiently small that they do not significantly effect the constraints. Further, since the SA channel is sub-dominant and the GCE preferred region is for m S ≈ m h , we neglect the effects of the different Higgs boost of eq. (3.5). This gives us the bound i,ann We show the resultant upper bound on λ HS in figures 3 and 4 in pink. We see that for J = 4, the best-fit region is not excluded. However, smaller values J 3 start to be in tension. This is in line with other studies that have found that the GCE and dSph bounds are consistent only for an enhanced signal from the galactic center [65].
Limits from collider searches and direct detection are independent of SA (and µ 3 ), and similar to bounds on for a scalar singlet Z 2 model. Collider searches in monojets and jets + MET set no current bounds in the mass range of interest, and are not expected to be constraining in the near future [45,67,68].
Direct detection bounds in contrast are very relevant. The spin-independent scattering cross section is 2) where f N is the Higgs-nucleon coupling 3) and m N = 0.946 GeV. We follow ref. [69] and take f N = 0.345 in placing our limits. See refs. [69][70][71][72][73][74][75][76][77][78] for more details. The strongest spin-independent scattering limits for DM masses 5 GeV are from the preliminary run at LUX [79]. For m S = 140 GeV, LUX excludes σ n > 1.7 × 10 −45 cm 2 . We plot the resultant upper limits in figures 3 and 4 in orange. For m S < m h , these are the most stringent limits on λ HS .

Conclusions
In this paper, we studied the simplest example of semi-annihilating DM, the Z3SSM, in the context of the Fermi GCE. The Z3SSM has only three parameters relevant to DM phenomenology; we reduce this to a two-dimensional parameter space by demanding the relic density reproduce observations. Our main result is given in figures 3 and 4, which JHEP06(2016)156 show the best-fit regions as well as constraints from direct detection, dwarf spheroidals and vacuum stability. We find that, assuming the signal from the galactic centre is at the higher range of expectations for an NFW profile, this model can explain the GCE while remaining consistent with all current constraints. The agreement with the measured spectral excess is quite good for a 20-bin analysis with χ 2 = 2.1 at the best fit point and spectrum. SA is a subdominant contribution to the γ-ray flux, but plays an essential role in obtaining the correct relic density and easing bounds from LUX and from dwarf spheroidals. This model can provide good fits to three of the different residual spectra presented by Fermi. The best fit is when the excess is modelled as a single power-law through out 20 energy bins, and interstellar gamma-ray emission is accounted for with intensity-scaled Pulsar or OB-star models; a weaker fit is possible for a spectra modelled with different power-law in each bin and an index-scaled Pulsar background. Other spectra presented by Fermi are either too hard to too soft to be reproduced.
Finally, we note that the region where this model explains the GCE only marginally evades the current bounds. Moderate improvements in either type of limit should exclude or, more optimistically, discover the region of parameter space discussed here. In particular, the full LUX results are expected to improve the direct detection bounds by a factor of ∼ 5 in the relevant mass range [80]. Thus we expect this explanation of the GCE to be proved or falfisied soon.

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was completed. YC was supported by the Australian Research Council. This work was supported by IBS under the project code, IBS-R018-D1.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.