A model explaining neutrino masses and the DAMPE cosmic ray electron excess

We propose a flavored $U(1)_{e\mu}$ neutrino mass and dark matter~(DM) model to explain the recent DArk Matter Particle Explorer (DAMPE) data, which feature an excess on the cosmic ray electron plus positron flux around 1.4 TeV. Only the first two lepton generations of the Standard Model are charged under the new $U(1)_{e\mu}$ gauge symmetry. A vector-like fermion $\psi$, which is our DM candidate, annihilates into $e^{\pm}$ and $\mu^{\pm}$ via the new gauge boson $Z'$ exchange and accounts for the DAMPE excess. We have found that the data favors a $\psi$ mass around 1.5~TeV and a $Z'$ mass around 2.6~TeV, which can potentially be probed by the next generation lepton colliders and DM direct detection experiments.


I. INTRODUCTION
The newly released data from the DArk Matter Particle Explorer (DAMPE [1]) exhibits an intriguing excess of the cosmic ray electron plus positron (hereafter CRE) flux at energies around 1.4 TeV [2]. We provide here a dark matter (DM) explanation based on a simple flavored U(1) extension of the standard model (SM). This kind of extension is known for quite a while [3][4][5]. Well-studied scenarios are those involving the second and third generation, U(1) µτ (denoted as L µ − L τ in the literature), which are partially motivated by the large mixing angle inferred from atmospheric neutrino oscillations [6][7][8]. Such models are recently used to explain anomalies in Higgs and quark flavor physics (see, e.g. [9][10][11][12]). This class of models was also discussed in the context of the PAMELA, ATIC and FERMI results [13].
In this work, we focus on another variant, U(1) eµ , under which only the first two generation leptons are charged. This choice is inspired by DAMPE CRE data as we are trying to establish the connection between the DM explanation for the CRE excess and the neutrino mass generation mechanism. In this framework, the DM candidate is a vector-like fermion ψ whose stability is guaranteed by an accidental U(1) symmetry. The DM annihilation into e ± and µ ± (also ν e and ν µ ) can account well for the DAMPE excess. Since the generated electrons and positrons lose energies quickly on the way to the Earth, the CREs detected by DAMPE must come from regions close to the solar neighborhood. As a result, we assume that there exists a nearby DM subhalo, which is also predicted by the structure formation of the cold DM scenario (e.g. [14,15]).

II. THE MODEL
Our model is a rather minimal extension of the SM. We add one additional anomaly-free U(1) eµ gauge group, two additional scalars, φ 1 and φ 2 , whose vacuum expectation values (vevs) break the new U(1) eµ spontaneously, three right-handed neutrinos, and a vector-like fermion ψ as a DM candidate. Only the lepton doublets, right-handed leptons and neutrinos of the first two Field L e L µ e R µ R N 1 N 2 N 3 ψ φ 1 φ 2 U(1) eµ charge 1 −1 1 −1 1 −1 0 q ψ 1 2 generations are charged under U(1) eµ as summarized in Table I. The fermion ψ is stable since the Lagrangian carries an additional accidental U(1) symmetry which can be interpreted as ψ-number.
In this model, the U(1) eµ symmetry demands both the charged lepton and the neutrino Yukawa couplings to be diagonal in the flavor basis. On the other hand, when the scalars receive a vev the resulting right-handed neutrino mass matrix is an unconstrained symmetric matrix: where y i j are Yukawa couplings of the right-handed neutrinos with the scalar singlets φ 1 and φ 2 , and M 12 and M 3 are mass parameters. With such structures we can reproduce the neutrino masses and mixing angles via the Type-I seesaw mechanism. The scalar potential in the unbroken phase reads where H is the usual SM Higgs doublet, and we have µ 2 H > 0 and µ 2 φ i > 0 for i = 1, 2. After electroweak and U(1) eµ symmetry breaking, 2, there exist three physical CP even Higgs bosons h and η i with masses m h and m η i , and one CP odd Higgs boson ζ with a mass m ζ . For simplicity, we assume here that the κ i are negligibly small so that h is identified with the SM Higgs boson. The κ i terms could be probed with future Higgs precision data. A careful and detailed study is, however, beyond the scope of this work.
The mass of the new gauge boson is m 2 Z = g 2 eµ (v 2 1 + 4v 2 2 ) on tree level, where g eµ is the U(1) eµ gauge coupling. Since the φ i do not carry any SM quantum numbers, the masses of the SM gauge bosons are not affected by φ i on tree level.
The relevant Lagrangian for the DM annihilation into SM fermions f is where q f labels the U(1) eµ charge of the field f , c.f. Table I. The SM fermion masses are neglected due to m f m ψ for our regions of interest. We further assume that the extra scalars, η i and ζ, and the right-handed neutrinos are all heavier than ψ and Z .
The DM annihilation cross-section into a SM fermion pairf f , σ(ψψ → Z →f f ), multiplied by the DM relative velocity v rel , is where c f =1 (1/2) for e and µ (ν e and ν µ ), and s = 16 m 2 ψ /(4 − v 2 rel ) is the square of the center-of-mass energy. Note that σv rel is dominated by the s-wave component as v rel → 0. The total Z decay width intof f andψψ reads

III. PARAMETER SPACE
We first study the CRE background, i.e., the CRE not from DM annihilations. The background component (from astrophysical sources such as supernova remnants and/or pulsars) is assumed to have a double-broken power-law form as with the first break at E br,1 ∼ 50 GeV and the second one at E br,2 ∼ 900 GeV according to the Fermi-LAT [16] and DAMPE observations [2]. During the analysis, we fix E br,1 to 50 GeV, and the sharpness parameter δ to 10. The fit to the DAMPE data with the e ± energy between 25 GeV and 4.6 TeV without taking into account the peak (excess) leads to Φ 0 = 247.2 GeV −1 m −2 s −1 sr −1 , γ = 3.092, ∆γ 1 = 0.096, ∆γ 2 = −0.968, and E br,2 = 885.4 GeV. Next, we include the contribution from a nearby DM subhalo in addition to the background and fit again to the data. The density distribution inside the subhalo is assumed to be a Navarro-Frenk-White profile [17], with a truncation at the tidal radius r t [18]. For the determination of the density profile of the subhalo, we refer to Ref. [19]. As for the propagation of electrons and positrons in the Milky Way, we adopt the Green's function approach presented in Ref. [20].
The background parameters E br,2 and ∆γ 2 are correlated to the DM component, and thus are being varied in the fit. Other parameters are fixed to the best-fit values obtained in the aforementioned background-only fit. Fig. 1 shows the model prediction of the CRE flux for m ψ = 1.54 TeV, σv rel = 6.82 × 10 −25 cm 3 s −1 , and the DM subhalo with a mass of M sub = 1.25 × 10 6 M at a distance of d = 0.1 kpc from the Earth.
There are four relevant DM parameters in this model: m ψ , m Z , g eµ , and q ψ . To ensure the DM model withstands various experimental bounds and to explore favored regions of the parameter space, we consider the constraints from (i) the relic density, (ii) the cosmic microwave background (CMB), (iii) the LEP measurements on the cross-sections of the leptonic final states, (iv) DM direct detection, and (v) the DAMPE data. Note that the recent measurements of the CRE flux by the Calorimetric Electron Telescope (CALET) up to 3 TeV [21] are not considered here, because of the relatively large statistical and systematic uncertainties. For the relic density, we use the Planck result: Ω ψ h 2 = 0.1199 ± 0.0027 [22] plus 10% theoretical uncertainties, which are commonly included to take into account the discrepancies among the different Boltzmann equation solvers and entropy tables.
The constraints on the DM annihilation rate from the PLANCK TT,TE,EE+lowP power spectra (Table 6 of Ref. [22]) are employed. Moreover, the LEP measurements on the cross-section of e + e − → + − can be translated into constraints on the new physics scale in the context of the effective four-fermion interactions [23] where δ = 0 (1) for f e ( f = e), and η i j = 1 (−1) corresponds to constructive (destructive) interference between the SM and new physics processes. For e + e − → e + e − (e + e − → µ + µ − ), one has Λ = 18 (21.7) TeV, which implies m Z /g eµ 7.2 (6.1) TeV.
Even if DM couples only to leptons at tree level, spin-independent DM-proton interactions can still be loop-induced and probed as discussed in, for instance, Refs. [24][25][26]. A recent updated analysis based on a leptophilic dark sector in Ref. [26] attains the constraints from direct detection on the DM and mediator mass for different types of DM-lepton interactions, as displayed in Fig. 2 therein. To apply the results to our model, we take the direct detection constraints in Ref. [26] for the vectortype interaction (solid blue line in their upper-right panel of Fig. 2) and then rescale it with our coupling constants. To realize our U(1) eµ model, only the vector couplings for e and µ are nonzero, g Ve , g Vµ 0 in the notation of Ref. [26]. Furthermore, the direct detection limit given in Ref. [26] is based on the LUX WS2014-16 run [27] which is slightly less stringent than that from the latest PandaX-II data [28]. As a consequence, with the new data the lower bound on the mediator mass will improve by a factor of [σ SI χp (LUX)/σ SI χp (PandaX)] 1/4 , given a DM mass. With these rescalings taken into account, the derived bound for m Z ∼ O(TeV) is where we set q e,µ = 1. The XENON1T [29] data yield a similar limit. The DM particle mass m ψ in the analysis ranges from 0.5 to 5.0 TeV with the Z mass in the range m ψ < m Z < 2m ψ , making the current σv rel larger than it was at the time of DM freeze-out, although the resonance enhancement needs not to be enormous. The DM charge q ψ is varied between 0.5 and 5. We conducted a random scan and a Nest-Sampling scan of the parameter space. After identifying the high probability region by checking the result of the random scan, we utilized MultiNest [30] in the Nest-Sampling scan to optimize the coverage of sampling. The two scans (∼ 10 8 points) are then combined with a profile likelihood method.
In Fig. 2 and Fig. 3, we present the 68% (inner) and 95% (outer) profile likelihood contours on the plane of m ψ − σv rel and m ψ − m Z , respectively. The preferred DM mass region is between 1.4 TeV and 1.7 TeV with a Z mass between 1.9 TeV and 3.2 TeV and g eµ between 0.014 and 0.38 at the 95% CL. We find no preferred region for q ψ over the scan range [0. 5,5].
Together with the coupling limits from PLANCK (relic density and CMB), the DM annihilation cross-section is confined within [3 × 10 −26 , 3 × 10 −24 ] cm 3 s −1 as shown in Fig. 2. The annihilation cross-section is inversely proportional to the mass of the subhalo, which is restricted inside the range of [2.5 × 10 5 , 6 × 10 7 ] M , assuming a distance of d = 0.1 kpc. For different values of d, the required subhalo mass scales approximately as d 2 [19].

IV. OTHER CONSTRAINTS AND PROSPECTS
We further consider bounds from DM indirect detection, and also comment on the model's detectability at future DM direct detection and lepton colliders.
• Fermi-LAT γ-ray data We have checked that the inverse Compton emission from the diffuse electrons and positrons for the presumed subhalo is negligibly small. On the other hand, we also study the γ-ray emission produced via the internal bremsstrahlung process by the charged fermions which come from the Z decays. The process is known as the final state radiation (FSR; [31]). The FSR γ-rays from the subhalo are essentially extended over a considerable patch of the sky. The expected numbers of photons from the DM annihilation within the subhalo for E γ > 100 GeV are estimated to be 0.7, 2.0, 5.9, 13.6, 25.3, 34.1, 34.4, for the integral radius of 0.1 • , 0.3 • , 1 • , 3 • , 10 • , 90 • , 180 • respectively around the halo center, assuming an exposure of 3 × 10 11 cm 2 s for ten years of operation of the Fermi-LAT. The corresponding numbers of the extragalactic background photon emission, according to the Fermi-LAT measurements [32], are 0.001, 0.01, 0.1, 1.1, 11.8, 776.6, and 1553.2, respectively. If the center of the subhalo is located in the inner Galaxy direction, the corresponding diffuse background could be higher by 10 − 100 times [33]. It implies that the detection of the γ-ray emission from such a subhalo is challenging (and hence unconstrained) to the Fermi-LAT in light of the small number of photons and a very long exposure time. The future ground based Cherenkov Telescope Array (CTA; [34]) may be able to detect such an extended γ-ray source and test our model. The Fermi-LAT γ-ray observations of the Milky Way halo set an upper limit of σv rel 5 × 10 −24 cm 3 s −1 for m ψ ∼ 1.5 TeV, presuming Majorana DM which annihilates into µ + µ − only. The DAMPE-favored parameter region is completely free from this constraint.
• IceCube ν data The IceCube observations of neutrinos from the Galactic center region give upper limits on the DM annihilation crosssections (again assuming Majorana DM) of 9.6 × 10 −23 cm 3 s −1 and 2.6 × 10 −22 cm 3 s −1 for the µ + µ − and νν channels, respectively [35]. These values are much larger than what is required to explain the DAMPE data, and no constraints can be imposed on our model from the Galactic center neutrinos. On the other hand, the subhalo itself may also be visible to IceCube. The DM annihilation rate within the halo can be characterized by Q = ρ 2 dl dΩ, where ρ is the density distribution, l is the line-of-sight path length, and Ω is the integral solid angle. The annihilation rate of the subhalo for an opening angle cone of 10 • is around two times higher than that of the Galactic center. It implies the previous bounds on the cross-sections will be improved by a factor of 2 in the presence of the subhalo. The favored region is, however, far below the new bounds. All in all, the current IceCube sensitivity is not able to constrain the parameter region yet.
• LZ sensitivity As shown in the previous section, the preferred regions to account for the DAMPE bump and to reproduce the correct relic density are centered around m Z ∼ 2.6 TeV with g eµ √ q ψ ∼ 0.1. Therefore, a large part of parameter space is unaffected by the PandaX-II search. The next generation DM experiment LUX-ZEPLIN (LZ) [36,37], however, can further improve the bound on the DM-nucleon cross-section by a factor of 50 or so, i.e., σ SI χp ∼ 2.4 × 10 −11 pb for TeV DM, before reaching the neutrino floor. It implies as indicated by the red contours in the Fig. 3. In other words, the LZ can probe a sizable part of the preferred region.
• ILC sensitivity The LEP measurements on e + e − → e + e − , µ + µ − require the effective scale of new physics Λ (which contributes to these processes) to be above 20 TeV. Future e + e − colliders, such as ILC [38], FCC-ee (formerly known as TLEP [39]) and CEPC [40], can further improve the limit. The ILC, for instance, with an integrated luminosity of 1000 fb −1 , can probe the new physics scale Λ beyond 75 TeV [38,41] via the process e + e − → µ + µ − , leading to the bound m Z /g eµ 21 TeV. The precise value of the lower bound depends on systematic uncertainties and the polarization of the electron and positron beams at the ILC.
As shown in Fig. 3, the combination of ILC and LZ projected sensitivities can disfavor a large region of the parameter space. Assuming ILC and LZ find no evidence of new physics, only the resonance region (2m ψ ≈ m Z ) remains viable.

V. CONCLUSION
In this work, we propose a simple U(1) eµ flavored neutrino mass model inspired by the DAMPE e + + e − excess at energies around 1.4 TeV [2]. The first two generations of leptons are charged under U(1) eµ while the third one is neutral. After U(1) eµ and electroweak symmetry breaking, the right-handed neutrino Majorana mass matrix is featureless, while the neutrino Dirac mass matrix is diagonal in the flavor basis. The observed neutrino masses and mixing angles can hence be easily realized via the Type-I seesaw mechanism.
The DM particle, a U(1) eµ -charged vector fermion ψ, annihilates into electrons, muons and neutrinos. To account for the DAMPE excess, a local DM subhalo with a mass of M sub = 1.25 × 10 6 M at a distance of 0.1 kpc from the Earth is needed. CREs lose energy so quickly on the way towards the Earth that they mostly have to come from a nearby area. The preferred parameter region is centered around (m ψ , m Z ) ∼ (1.5, 2.6) TeV with σv rel ∼ 10 −25 cm 3 s −1 . We have scrutinized constraints from indirect searches (Fermi-LAT and IceCube), direct DM searches and LEP. Interestingly, a significant portion of the preferred parameter space is within the reach of the next generation lepton colliders and DM direct detection experiments.