Gamma Lines without a Continuum: Thermal Models for the Fermi-LAT 130 GeV Gamma Line

Recent claims of a line in the Fermi-LAT photon spectrum at 130 GeV are suggestive of dark matter annihilation in the galactic center and other dark matter-dominated regions. If the Fermi feature is indeed due to dark matter annihilation, the best-fit line cross-section, together with the lack of any corresponding excess in continuum photons, poses an interesting puzzle for models of thermal dark matter: the line cross-section is too large to be generated radiatively from open Standard Model annihilation modes, and too small to provide efficient dark matter annihilation in the early universe. We discuss two mechanisms to solve this puzzle and illustrate each with a simple reference model in which the dominant dark matter annihilation channel is photonic final states. The first mechanism we employ is resonant annihilation, which enhances the annihilation cross-section during freezeout and allows for a sufficiently large present-day annihilation cross section. Second, we consider cascade annihilation, with a hierarchy between p-wave and s-wave processes. Both mechanisms require mass near-degeneracies and predict states with masses closely related to the dark matter mass; resonant freezeout in addition requires new charged particles at the TeV scale.


Introduction
The existence of dark matter (DM) is one of the strongest pieces of evidence for physics beyond the Standard Model. Searches in cosmic rays for evidence of DM annihilation or decays are a cornerstone of the experimental effort to detect DM. Monochromatic photon lines, though in most models a subdominant signal, provide a particularly clean astrophysical signal [1,2,3,4,5].
Several recent analyses have claimed evidence for a distinct spectral feature in the Fermi-Large Area Telescope (LAT) [6] photon spectrum at around 130 GeV [7,8,9,10], in regions near the galactic center. Evidence for this feature has also been reported in galactic clusters [11] and in non-associated sources [12], although the latter claim remains contentious [13,14,15]. While the statistics are limited, the morphology of the signal may favor an explanation as annihilating DM [16,17,11]. The presence of an additional photon line at approximately 111 GeV [10,12] is highly suggestive, if also statistically limited, and would lend more credence to a particle physics explanation [18]. The Fermi collaboration's own search for photon lines uses slightly different search regions and methodology and sets an upper limit marginally in conflict with the claimed signal [19].
It remains to be established whether the excess is instrumental, astrophysical, or representative of an overly optimistic characterization of the systematic uncertainties in the galactic background [20,21]. However, standard thermal WIMPs are not capable of explaining the Fermi signal, and it is of interest to work out the necessary structure in DM models which could give rise to the 130 GeV line. Any dark matter model for the Fermi 130 GeV line must account for two interesting facts. First, there is no evidence for an excess in the continuum photon spectrum, which strongly constrains DM annihilation into the usual SM annihilation channels (ff , W + W − , ZZ) [17,22,23,24] or indeed into any charged final states. Since in most thermal models DM-photon couplings are generated radiatively from the DM couplings to charged final states [25,26], the typical line cross section is generically related to the cross-section for annihilation into charged modes X, X † by As the fragmentation and decay of the final states X, X † give rise to a continuum photon spectrum This brings us to the second interesting fact. The best fit cross-sections for the 130 GeV feature [7,8] are more than an order of magnitude smaller than the expectation for (s-wave) thermal freezeout. If annihilation to SM or charged modes is absent or suppressed as the continuum limits suggest, reconciling this inefficient annihilation to photons with the WMAP relic density Ω χ h 2 = 0.1109 ± 0.0056 [27] requires either: (1) a nonthermal relic abundance [28,29]; (2) non-photonic annihilation modes which are suppressed, possibly only post-freezeout, relative to the naive radiative scaling of Eq. (1) [30,29]; or (3) a mechanism to enhance the thermal annihilation cross-section to photonic final states in the early universe [31].
In the present work we will study two mechanisms which give enhanced DM annihilation to photons while also obtaining the correct thermal relic abundance, and build simple reference models for both.
Our first example is resonant freezeout, where the presence of a resonance in the DM sector spectrum enhances the annihilation cross-section into photons during freezeout. Our second example introduces an intermediate annihilation mode, so that DM annihilation proceeds through a cascade decay of a non-photonic intermediate state,χχ → φφ ′ → 4γ. In this example, the relation between the crosssection necessary for thermal freezeout and the present-day gamma line cross section is explained by the interplay of s-wave and p-wave contributions to the annihilation.
In section 2, we perform an effective operator analysis of DM-photon couplings and demonstrate the need for new particles at mass scales comparable to the DM mass. In section 3 we perform a detailed examination of resonant freezeout, and consider cascade decays in section 4. Section 5 contains our conclusions.

Effective Operators for DM Annihilation to Photons
To demonstrate the need for multiple new states in thermal models for the Fermi 130 GeV line, we begin by writing effective operators to describe the interaction between DM and one or two photons. We assume an unbroken Z 2 symmetry to explain the stability of the DM particle, and for simplicity work after electroweak symmetry breaking. We first consider the case when DM is a (Dirac) fermion. The leading operators coupling DM to photons are the electric and magnetic dipole operators, χσ µν χF µν and χσ µν χF µν , appearing at dimension five [32,33]. However, the dominant annihilation process mediated by these two operators is χχ → ff through a photon in the s-channel [30]. This problematic annihilation to ff can be suppressed in the present day if the Dirac fermion is split into two Majorana χ 1,2 , with a small mass gap m 2 −m 1 such that the depleted abundance of the heavier χ 2 post-freezeout shuts off the charged annihilation channel [30,29]; in this case, the DM is the Majorana χ 1 and the EFT containing only χ 1 is indeed insufficient to describe the freezeout.
At dimension six, there are four operators which couple DM pairs to two photons, two CP-conserving operators and two CP-violating operatorsc In addition there are operators with a single photon, such as χχF µν Z µν , χγ 5 χF µνZ µν , and χγ µ χF µν ∂ ν h.
We will concentrate on the operators with two photons, but we will comment on other operators when we introduce a concrete UV model.
For scalar DM, the first DM-photon interactions appear at dimension 6, For simplicity we have taken φ complex, but this is only necessary for O sc,1 . As for the fermionic dipole operators, O sc,1 will dominantly mediate annihilation to ff, which can be suppressed by a mass splitting between real and imaginary parts of φ.
For fermionic DM, the presence of independent operators which contribute in different leading partial waves to the DM annihilation cross-section show that it is easy to accommodate an apparent suppression in the DM annihilation cross-section. In an EFT consisting of the CP-conserving operators in Eq. (2), the cross-section is For a mixed partial wave freezeout process, the dark matter relic abundance is approximately given by [34] in terms of the freeze-out temperature In Fig. 1 we show in the left panel the region of s and p giving the correct relic density, and in the right panel translate that into the region of operator coefficients c 1 /Λ 3 and c 5 /Λ 3 . From Fig. 1, we can see that to simultaneously accommodate the thermal relic abundance and the best-fit cross section for  the natural magnitude of c 1 and c 5 is α/π, so without additional enhancement factors, the necessary cutoff Λ is much smaller than m χ , invalidating the EFT.
In the next two sections, we study two models which extend some of the effective operators of in Eq. (2) and Eq. (3) by introducing a (pseudo-)scalar degree of freedom φ which couples to hypercharge gauge bosons through a loop of charged fermions, where Y is the hypercharge of the new charged fermions f i under U (1) Y , and i = 1 · · · N f is the flavor index of the N f fermions. We integrate out f i to generate the following effective operators [35] N Here we have absorbed the electric charge e f , the Yukawa coupling y f and the multiplicity factor n f into the definition ofm f 1 . There are two more operators F µν Z µν and Z µν Z µν generated. They provide a sub-leading contribution to the dark matter annihilation cross section because of the θ 2 W ∼ 0.05 suppression factor. We neglect these operators in our analysis. We also point out that if the ∼ 114 GeV gamma line becomes robust, our analysis should be extended to include operators constructed from W a µν to fit the relative fluxes inχχ → γγ andχχ → γZ.

Resonant Freezeout
One way to enhance the dark matter annihilation cross section at freezeout relative to the cross-section today is by introducing a resonance with mass slightly above twice the DM mass, for 0 < δ ≪ 1 [31,36]. In this section we will discuss the parameter space for a resonant freezeout model where DM annihilation to photon pairs is entirely responsible for setting the thermal abundance.
To be concrete, we consider a simple UV completion of the EFT in the previous section, namely Dirac fermionic dark matter χ coupling to photons through a pseudo-scalar a with We take the loop-induced aF F coupling to be given by a loop of charged fermions, as shown in Fig. 2, so the effective coupling is where y f is the Yukawa coupling of a to the heavy fermions f whose mass, number, and charge are given by m f , N f , and e f .

s-wave
The operators in Eq. (11) yield the annihilation cross section where Γ a is the total width of a, γ ≡ Γ a /m a , and in the second line we have taken v, δ, γ ≪ 1 [31].
We can write the thermally averaged cross-section as where is the cross-section at x → ∞, and contains the information about the nontrivial velocity dependence of the cross-section. The width of the pseudo-scalar is bounded from below by its couplings to dark matter (in the relevant parameter space, the radiative width into photons is negligible). For δ ≪ 1, We first establish that our model has a reasonable range of parameter space which can give a sufficiently large cross-section for χχ → γγ.
where we have used the Einasto value for σv 0 and dropped terms of order γ 2 . This determinesm f as a function of δ and λ i χ , showing that obtaining the present-day cross-section requires δ 0.1. After using the present-day best fit cross-section to fixm f in terms of λ i χ , δ, the thermal freezeout is controlled by δ and the minimum allowable width, as determined by Eq. (17). Therefore if γ is too small, e.g. λ i χ ≤ 0.4 in Fig. 3, the annihilation is too efficient and the yield is too small to account for the present-day dark matter abundance in the absence of other decay modes for a. If, on the other hand, γ is too large, e.g. λ i χ ≥ 0.6 in Fig. 3, annihilation is inefficient and could even over-close the universe. Annihilation becomes less efficient both gradually at large δ and rapidly at very small δ, as the pole passes outside the main peak of the Maxwell-Boltzmann distribution during freezeout.
In Fig. 4 we show the contours in the δ-λ i χ plane yielding Ω χ h 2 = 0.11. To illustrate the astrophysical uncertainties, we quote results for both the Einasto and NFW gamma line best-fit cross sections. Comparing the resulting contours, we can see that the astrophysical uncertainty introduces an order 50% uncertainty on the couplings of the pseudo-scalar to dark matter. In the green dashed line of Fig. 4, we show the contour for the case where a has an additional decay mode XX † with Br(a → XX † ) = 10 Br(a → γγ).
As can be seen from Fig. 4, there is no real upper bound on the (scaled) fermion massm f . However, from Eq. (18),m f is inversely proportional to δ, to obtain the present-day annihilation cross section. If one requires δ > 0.01 from considerations of fine-tuning, the scale for the charged fermion masses should be bounded from above by around 500 GeV, or in other words, within collider-accessible energies.

s + p-wave
We now consider the case where more than one partial wave is important for freezeout. For concreteness we extend the single pseudo-scalar model of the previous subsection to include a CP-violating coupling λ r χ aχχ to the dark matter, Pure p-wave freezeout would require extremely tuned values of the resonance mass, δ ∼ 10 −5 , to obtain the present-day observed cross-section. In the more interesting case of mixed s and p-wave resonant freezeout, the present-day observed cross-section, and hence the necessary values of δ, are set by the s-wave contribution, but p-wave scattering can dominate the annihilation at freezeout.
When p-wave scattering dominates freezeout, the additional v 2 dependence of the cross-section shifts the velocity integral to higher values, reducing the overlap with the pole; narrower resonances are therefore required to achieve cross-sections sufficiently efficient to yield the observed relic abundance.
Therefore when p-wave scattering dominates during freezeout, additional contributions to the a width are more constrained than in the s-wave case. Since we will be interested in cases with a hierarchy between p-wave and s-wave modes, it is important to include the contribution to γ from λ r χ , γ r = δ 3/2 λ r 2  As for the pure s-wave case, we show the contours of Ω χ h 2 = 0.11 in the δ-λ i χ andm f -λ i χ planes in Fig. 5. We plot three different ratios of λ r χ /λ i χ . For λ i χ λ r χ √ δ, the contribution to the resonance width from λ r χ is negligible. The additional p-wave contribution to the freeze-out cross section prefers a smaller value of λ i χ , which explains why the pure s-wave contour (red, solid) in the left panel lies at larger values of λ i χ than the blue dashed contour, which denotes a case where p-wave scattering is important for annihilation but not for the resonant width. For λ i χ λ r χ √ δ, the contribution to the resonance width from λ r χ is non-negligible, reducing the enhancement from the resonance. As a result, larger values of λ i χ are required to obtain sufficiently efficient annihilation, as can be seen from the green, dotted contour in the left panel.
As can been from the right panel of Fig. 5, the photon vertex scalem f is again below around 500 GeV for δ 0.01, implying collider-accessible charged particles.  [37,38,39] and applied to the Fermi-LAT gamma line excess in [40,9]. Here, we write down a simple explicit model and discuss the parameter space for realizing a thermal relic abundance together with an explanation for the Fermi-LAT gamma line excess.
As before, we consider Dirac dark matter χ, together with a real scalar field which we denote φ, and study the following set of interactions Here, as before, the scalem f ≡ 2N f y f e 2 f /3m f arises from some heavy charged fermions that are integrated out to generate the φF F operator; we require only that m f is sufficiently large to forbid DM from annihilating into charged fermions. We are interested in the parameter space where m φ < m χ , so the dominant annihilation channel for DM is χχ → φφ → 4γ, as shown in Fig. 6.
The energies of the two photons from each φ decay are, in the dark matter rest frame, where θ is the angle between the photon direction in the φ rest frame and the φ direction of motion.
Because φ is a scalar field, the distribution is isotropic in θ, and the photon spectrum is evenly distributed between the kinematic endpoints: which, as ǫ → 0, limits to a delta function centered at m χ /2. Here Θ(x) is the usual Heavyside function.  Since the gamma line spectrum can provide a good fit to Fermi LAT data, we also anticipate a good fit for this model for sufficiently small ǫ. To estimate upper bounds for ǫ, we consider the average Fermi-LAT energy resolution for gamma ray energies above 50 GeV, σ(E)/E ≈ 0.10 + 0.0001 E/GeV To work out the parameter space for this model we translate the best-fit line spectra of [8]   Our explicit model considers fermionic DM and, for economy, a single real scalar φ. Thus parity dictates that the annihilation is p-wave suppressed unless both λ r χ and the CP-violating λ i χ are nonzero. In terms of the mass splitting ǫ, the cross-section for χχ → φφ is where we have kept the leading terms in the limit ǫ ≪ 1.
In Fig. 8, we show the contours in the λ i χ -λ r χ plane which give the required dark matter annihilation cross section for explaining the Fermi-LAT gamma line excess, for m χ = 260 GeV and ǫ = 0.02. The points which give the relic abundance Ω χ h 2 = 0.11 are indicated by heavy circles. Hierarchies of order ∼ 10 between CP-preserving and CP-violating couplings are required, suggesting small (but not tiny) CP violation in a DM sector. We also show the case where the branching fraction of φ into two photons is 20%.
An equally well-motivated alternative to the CP-violating reference model considered here would be to allow the Dirac DM χ to annihilate to a nearly degenerate pair of scalars with opposite parity, φ and a, with subsequent decays to photons. In this case, the s-wave annihilation χχ → φ a is proportional to (λ φ χ λ a χ ) 2 , while the p-wave annihilation is proportional to ∼ (λ φ χ ) 4 + (λ a χ ) 4 . If the branching fractions of φ, a into photons are order 1, accommodating the dark matter thermal relic abundance and the present-day annihilation cross section for the Fermi-LAT gamma line still requires a factor of ∼ 10 hierarchy between λ φ χ and λ a χ .

Discussion and Conclusions
We have explored the possibilities for thermal models for the Fermi-LAT 130 GeV gamma line excess where the dominant DM annihilation channel is into photons. We consider two mechanisms, (1) models where a resonance in the DM spectrum enhances the annihilation rate during freezeout and to a lesser extent at the present day, and (2) models where DM annihilates to a new intermediate state which subsequently decays to photon pairs. Here the interplay of the s-wave and p-wave annihilation is responsible for reconciling the necessary cross-section at freezeout with the observed cross-section today. For the resonance model, charged fermions at the TeV scale are predicted and are accessible at the Large Hadron Collider, which could be interesting if the excess in the Higgs diphoton decay channel persists [42,43].
Both of the classes of models considered in this paper require coincidences in the mass spectrum.
For resonant freezeout, the resonance mass m a must be within a percent of twice the dark matter mass, a striking coincidence. For cascade decays, the intermediate state(s) φ must be within again about a percent of the dark matter mass in order to have a sharp enough spectral feature to fit the data well. These near-degeneracies are suggestive of a dark sector with one scale Λ m setting the overall mass scale, and another Λ G ≪ Λ m determining splittings. Composite dark sectors are an appealing avenue to flesh out the models we have considered. Near-degeneracies in the dark sector spectrum, as necessary for the cascade annihilation model, are readily accommodated in composite sectors. Resonances above threshold pose a somewhat more complicated picture. As we know from heavy quarkonia in the SM, a spectrum with resonances slightly above threshold could certainly be obtained; the complication is that such resonances would necessarily be accompanied by a resonance below threshold, giving the thermal relic abundance a detailed dependence on the parameters of the model. We leave detailed model building to future work.