Explain DAMPE Results by Dark Matter With Hierarchical Lepton-Specific Yukawa Interactions

We propose to interpret the DAMPE electron excess at 1.5 TeV through scalar or Dirac fermion dark matter (DM) annihilation with doubly charged scalar mediators that have lepton-specific Yukawa couplings. Hierarchy of such lepton-specific Yukawa couplings is generated through the Froggatt-Nielsen mechanism, so that the dark matter annihilation products can be dominantly electrons. Stringent constraints from LEP2 on intermediate vector boson production can be evaded in our scenarios. In the case of scalar DM, we discuss one scenario with DM annihilating directly to leptons and the other scenario with DM annihilating to scalar mediators followed by their decays. We also discuss the Breit-Wigner resonant enhancement and the Sommerfeld enhancement in case that the s-wave annihilation process is small or helicity suppressed. With both types of enhancement, constraints on the parameters can be relaxed and new ways for model building will be open in explaining the DAMPE results.


Introduction
The nature of dark matter (DM) is one of the most important questions in particle physics and cosmology. In fact, many new physics theories beyond the standard model (SM) can provide viable DM candidates. Apart from its various gravitational influences, dark matter has so far eluded all direct detection experiments through nucleon recoil, prompting ones to find alternative ways to search for its existence. An important method of probing DM properties is via indirect detections, whereby we look for the appearance of particles like high energy gamma-rays, positrons or X-rays produced via annihilation or decay of DM concentration in galaxies (satellites, dwarfs, or clusters).
In past few years, positron excess has been reported by various experiments, such as AMS02 [1,2], PAMELA [3,4] and Fermi [5]. Recently, the DArk Matter Particle Explorer (DAMPE) satellite [6], which is a new cosmic ray detector with excellent energy resolution and hadron rejection power, published their measurements about the cosmic e + + e − flux up to 5 TeV and announced to find a sharp peak at ∼ 1.5TeV. Although both astrophysical (e.g., pulsars) and DM origins are possible to act as the required nearby mono-energetic electrons sources, the DM explanation could potentially guide the search of DM particles in future direct detection and collider experiments. Many DM models had been proposed to explain the DAMPE results . In the DM explanation (DM annihilation into electrons or equal amounts of lepton flavors), the best fit value of the DM mass should be 1.5 TeV if the nearby DM sub-halo locates at 0.1 ∼ 0.3kpc away form the solar system and the DM annihilation cross section should be σv ≈ 3 × 10 −26 cm 3 /s. Besides, in order to guarantee the resulting electron/positron spectrum to be a narrow peak instead of a box shape spectrum, the required mass ratio between the mediator (to lepton pairs) and the DM mass is stringently constrained to be higher than 0.995 by numerical fitting to the DAMPE data [13].
In order to have a large DM annihilation cross section in the DM sub-halo and at the same time give the correct DM relic density, it is preferable to adopt Dirac DM scenarios because the Dirac DM annihilation into lepton pair final states via vector leoptophilic mediator will not be s-wave suppressed [36]. Scalar DM scenarios, which can annihilate into vector mediator pairs followed by their late-time decays, can also explain the result. Both scenarios tend to adopt a leptophilic gauge boson as the mediator which can couple universally to the lepton flavors. On the other hand, the DM direct detection bounds as well as the electron/positron collider constraints on the vector mediator production will impose rather stringent constraints on these scenarios.
An alternative possibility is the scalar-portal DM scenario, in which the mediator will mainly couple to leptons. We propose to explain the DAMPE excess by a scalar or Dirac fermion DM candidate with lepton-specific Yukawa couplings. A horizontal family symmetry U (1) H , which can be either global (with small explicit symmetry breaking terms) or local, is introduced for the lepton sector with Froggatt-Nielsen mechanism [37,38]. By properly choosing the U (1) H quantum numbers, the Yukawa couplings between the scalar mediators and the first family leptons can be unsuppressed while other types may be suppressed, which is just needed for explaining the DAMPE result.
In general, the models with s-wave suppressed DM annihilation cross section are not favored to interpret the DAMPE results. We note that, by adopting Breit-Wigner enhancement [39][40][41] or Sommerfeld enhancement [42][43][44], the s-wave (helicity or propagator) suppressed process can also be used to explain the DAMPE results because of a large enhancement factor. Therefore, new ways for model building in explaining the DAMPE results will be open with such an enhancement factor. This paper is organized as follows. In Section 2, we introduce a scalar DM model to explain the DAMPE electron/positron excess. Effects of Breit-Wigner enhancement and Sommerfeld enhancement are discussed. In Section 3, a model with Dirac fermion DM is proposed. Finally, we draw our conclusions in Section 4.

Scalar dark matter with scalar mediator involving lepton-specific interactions
We propose to explain the DAMPE result with scalar DM and scalar mediators. The complex scalar S, which is odd under a discrete Z 2 symmetry, will act as the DM candidate while other fields are even under Z 2 . A Higgs-portal type interactions between S and scalar mediator T , which will couple only to leptons, will be introduced. To generate the Yukawa hierarchy so that the mediator decays dominantly to electrons, a global U (1) H with small explicit symmetry breaking terms or an anomaly free local U (1) H horizontal symmetry will be introduced to generate the required suppression factors via Froggatt-Nielsen mechanism. The Lagrangian has the following form The non-renormalizable interactions involving the flavon field U , which transform nontrivially under U (1) H , are generated after integrating out the heavy modes at the scale Λ. Due to the charge assignments of U (1) H horizontal symmetry for the SM fermions and T , the Yukawa coupling of the formĒ c R,i E R,i T will appear only after the flavon acquires a VEV U .
From the U (1) H quantum numbers given in Table I, the Yukawa couplings between the scalar mediator and leptons have the following hierarchy , the resulting Yukawa couplings have the following hierarchy An alternative possibility is that we choose the following U (1) H quantum numbers The Yukawa couplings for e, µ are unsuppressed while for τ is suppressed, so y e ≃ y µ ≫ y τ . Similarly, we could have y e ≃ y τ ≫ y µ . For the DM annihilation, we have two possibilities • Scenario I: The DM particles annihilate into the scalar mediators SS * → T ++ T −− , followed by their decays into lepton pairs T −− → l R l R . So the dark matter mass should satisfy m χ ≡ m S > m T and the range of m T ∈ [0.995, 1] × m S to fit the peak shape eletron/positron spectrum required to fit the DAMPE data.
Our numerical results are shown in Fig.1. We can see that the Higgs portal coupling λ 2 is constrained to lie near 1.7, which is quite large but within the perturbative regime.
In previous papers with typical leptophilic U (1) gauge bosons as DM mediator [18,27], the LEP2 bounds on vector boson production will rule out many simplest scalar DM models. In our setting, the LEP2 constraints will be relaxed because the T scalar with double electric charge will not be produced directly on the electron/positron collier LEP2. So it is fairly advantageous to use such doubly charged scalar as the DM mediator.
Naively, the DM can scattering off the light quarks at one-loop order through T scalar loops that connect to the quark lines via photons. However, the one-loop diagram can be proved to vanish. So the leading order contributions arise at two loop order. The DM direct detection cross section are thus suppressed at two-loop order. By matching to the effective operator after integrating out the scalar T we can estimate the scalar DM-nucleon scattering amplitudes to be Following the methods in [45], we could estimate the DM direct detection cross section with the DM velocity v ∼ 10 −3 c and the velocity of the recoiled nucleus v d = 2E d /m N . With m T ∼ 3T eV , the DM-nucleus scattering cross section, which can be estimated to be of order 3 × 10 −16 pb, can easily survive the DM direct detection bounds given by LUX [46] and PandaX [47].
• Scenario II: The DM particles annihilate directly into four leptons S * S → (T ++ ) * (T −− ) * → l − l − l + l + if the scalar mediator is heavier than 3 TeV. Such annihilation cross section will in general be suppressed by the mediator propagator.
In order to enhance the DM annihilation cross section to explain the DAMPE data, we propose two ways in this scenario.
• Resonant annihilation with Breit-Wigner enhancement: If the scalar mediator mass lies slightly higher than the DM mass m S , the direct 4lepton final states annihilation will be enhanced by Breit-Wigner resonant effect [39].
The resonant annihilation cross section are given as (2.10) It is known that for γ ≪ 1 and γ 2 ≪ (v 2 rel − 2ǫ) 2 , we have the approximation We can see that we could have double enhancement near the threshold due to the apparent δ(0). We need to cut the apparent infinity with the decay width of T which is estimated to be given by So we have the cut for the delta function (2.13) Using the standard formula for DM relic density [41], Ωh 2 is given by So we can obtain a universal σv at the freeze-out time. The thermal averaged cross section today can also be readily obtained from J f with ǫ → 0. So, if this scalar DM scenario is to explain the DAMPE data, the range of λ 2 can be much smaller if y E is small because of the double enhancement effects near resonance. We should note that the resonant enhancement will affect the annihilation cross section at both the freeze-out temperature and today. The DM-nucleon scattering cross section will be suppressed by a tiny coupling λ 2 , so the DM direct detection experiments may not see any signal in the near future. • DM annihilation with Sommerfeld enhancement: There is an alternative possibility in which the DM annihilation in the early universe may be helicity suppressed (by light fermion mass) at the s-wave and the DM relic density is determined mainly by p-wave processes. Suppressed by the low relative velocity v ∼ 10 −3 c, the p-wave process will not give dominant contributions to the DM annihilation in the sub-halo. So, in order to explain the observed electron/positron excess reported by DAMPE, we need to enhance the suppressed s-wave contributions to the DM annihilation today.
Fortunately, the Sommerfeld enhancement has the desired behavior: it is enhanced at low velocities and therefore can boost the present day annihilation cross section. The DM annihilation in the early universe will be affected only slightly. We should note although chemical recoupling [44] implies suppressed indirect detection signals for near resonance regions, the Sommerfeld enhancement can still enhance the suppressed annihilation up to a factor 10 2 ∼ 10 3 to account for the DAMPE signal.
In this scenario, although the s-wave of DM annihilation is not suppressed, the requirement to give a correct DM relic density will stringently constrain the couplings involved. Considering the Sommerfeld enhancement can explain the DAMPE results with much smaller coupling strengths. Therefore, the DM direct detection bounds can be evaded very easily.
We can introduce an additional U (1) X gauge group with the gauge coupling strength g X and the gauge boson mass m X . The complex DM particle χ ≡ S will transform non-trivially under U (1) X with the corresponding charge Q S . The Sommerfeld enhancement factor can be approximated by [43] S ≈ g 2 X m S 4πm X . (2.16) for a massive gauge boson mediator A X µ . While the DM annihilation is dominantly by S * S → XX via t, u-channel S, the resulting cross section must give the correct DM relic density, which is given by the Planck satellite data Ω DM = 0.1199 ± 0.0027 [48] in combination with the WMAP data [49](with a 10% theoretical uncertainty). So we must have α X Y 2 S ≈ 0.149 for the DM mass m S = 3TeV. For simplicity, we choose the DM U (1) X charge Y S = 1. Numerical results on the value of Sommerfeld enhancement factor are given in Fig.2. We can see that, given an appropriate mass of the mediator φ, sufficient enhancement factor can be achieved at different cosmic objects, such as the dwarf halos, Milky Way and the clusters.
It is addressed in [44] that the DM can annihilate into A X µ φ with φ being the hidden Higgs that breaks the U (1) X gauge symmetry. The mass of φ is argued to be m φ 10m X by perturbative requirement. However, we can introduce additional Higgs portal terms, which can contribute to the mass of φ, so that the simple relation between m φ and m X is released. So we will neglect the effect of hidden Higgs φ in our estimation of σv rel by tuning φ to be very heavy. Openning both annihilation channels will introduce uncertainty for the parameter g X because of the free parameter m φ .
The Sommerfeld enhancement factor is determined by the DM wave-function obtained from non-relativistic Schrodinger equation: with which the approximate analytic results can be obtained after approximating the Yukawa potential by Hulthen potential [50]. The enhancement factor is thus approximated byS at the resonance and is proportional to v −1 away from resonance. In order to avoid the chemical recoupling, we will adopt mainly the v −1 enhancement behavior in our numerical study. In Fig.3 we can see that the coupling strength λ 2 can be reduced to a much smaller value with such an enhancement. As the DM annihilation cross sections into leptons are enhanced by the Sommerfeld factor, the reduced coupling strengths of λ will easily pass the DM direct detection constraints.

Dirac fermion dark matter with scalar mediator involving leptonspecific interactions
We also propose to explain the DAMPE results by introducing a Dirac DM scenario which can annihilate into leptons via a scalar portal. It is also possible to adopt the s-wave suppressed Majorana DM scenario with Sommerfeld enhancement and we will discuss that possibility in our subsequent studies.
The Lagrangian with Yukawa hierarchy from Froggatt-Nielsen mechanism is written as with Dirac DM χ and an additional real scalar S as a portal to doubly charged complex scalar T that couple to the lepton species. The discrete Z 2 parity is imposed only for the DM particle χ. Similar to the preceding section, we can have different annihilation modes, depending on the mass of S and T : • For m S ≃ 6 TeV and m T ∈ [0.995, 1] × m χ , the DM can annihilate directly into T T * pairs. This case is similar to the scenario I of scalar DM scenario. A new feature here is that resonant enhancement annihilation is possible when m S is very near 2m χ . The discussions are similar to that of scenario II in preceding section except that the m S can be slightly lighter than 2m χ , resulting in a negative ǫ parameter. In this case, the  reduced coupling strength λ can still be able to give the correct DM relic density and at the same time explain the DAMPE results. Numerical results on the parameters a 1 , y φ , which can account for both the DAMPE data and DM relic density, are given in Fig.4.
• For m S ≃ 6 TeV and m T m χ , the DM particles will annihilate directly into four leptonsχχ → S → (T ++ ) * (T −− ) * → l − l − l + l + . This case is similar to the correponding discussion in the preceding section. Both Sommerfeld enhancement and resonance can be used to avoid the propagator suppression. Due to the fact that three internal propagators exist, we could have triple enhancement due to the resonance. Sommerfeld enhancement are also possible by introducing an additional light scalar or gauge boson. So it is possible that the s-wave suppressed DM annihilation cross section could be enhanced enough to explain the DAMPE results. Besides, the reduced coupling strength λ can be also possible with a large ( mass dimension) trilinear coupling a 1 . Fig.5 shows the the scattering plot of σv with/without Sommerfeld enhancement in the case of Dirac fermion DM. The allowed range of a 1 versus the light gauge boson mass m φ with Sommerfeld enhancement is shown in Fig.6.

Conclusions
In this work we proposed to interpret the DAMPE electron excess at 1.5 TeV through scalar (or Dirac fermion) dark matter (DM) annihilation with doubly charged scalar mediators that have lepton-specific Yukawa couplings. Hierarchical lepton-specific Yukawa couplings are generated through Froggatt-Nielsen mechanism, so that the dark matter annihilation products can be dominantly electrons. Stringent constraints from LEP2 on intermediate vector boson production can be evaded in our scenarios. In the case of scalar DM, we discussed two scenarios: one scenario with DM annihilating directly to leptons and the other one with DM annihilating to scalar mediators followed by their decays. We also discussed the Breit-Wigner resonant enhancement and Sommerfeld enhancement in case that the s-wave annihilation process is small or helicity suppressed. With both types of enhancement, the constraints on the parameters can be relaxed and new ways for model building will be open in explaining the DAMPE results.