Radiative seesaw model and DAMPE excess from leptophilic gauge symmetry

The European Physical Journal C, Mar 2018

In the light of the \(e^{+}+e^{-}\) excess observed by DAMPE experiment, we propose an anomaly-free radiative seesaw model with an alternative leptophilic \(U(1)_X\) gauge symmetry. In the model, only right-handed leptons are charged under \(U(1)_X\) symmetry. The tiny Dirac neutrino masses are generated at one-loop level and charged leptons acquire masses though the type-I seesaw-like mechanism with heavy intermediate fermions. In order to cancel the anomaly, irrational \(U(1)_{X}\) charge numbers are assigned to some new particles. After the spontaneous breaking of \(U(1)_{X}\) symmetry, the dark \(Z_{2}\) symmetry could appear as a residual symmetry such that the stability of inert particles with irrational charge numbers are guaranteed, naturally leading to stable DM candidates. We show that the Dirac fermion DM contained in the model can explain the DAMPE excess. Meanwhile, experimental constraints from DM relic density, direct detection, LEP and anomalous magnetic moments are satisfied.

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Radiative seesaw model and DAMPE excess from leptophilic gauge symmetry

Eur. Phys. J. C Radiative seesaw model and DAMPE excess from leptophilic gauge symmetry Zhi-Long Han 2 Weijian Wang 1 Ran Ding 0 0 Center for High Energy Physics, Peking University , Beijing 100871 , China 1 Department of Physics, North China Electric Power University , Baoding 071003 , China 2 School of Physics and Technology, University of Jinan , Jinan 250022, Shandong , China In the light of the e+ + e− excess observed by DAMPE experiment, we propose an anomaly-free radiative seesaw model with an alternative leptophilic U (1)X gauge symmetry. In the model, only right-handed leptons are charged under U (1)X symmetry. The tiny Dirac neutrino masses are generated at one-loop level and charged leptons acquire masses though the type-I seesaw-like mechanism with heavy intermediate fermions. In order to cancel the anomaly, irrational U (1)X charge numbers are assigned to some new particles. After the spontaneous breaking of U (1)X symmetry, the dark Z2 symmetry could appear as a residual symmetry such that the stability of inert particles with irrational charge numbers are guaranteed, naturally leading to stable DM candidates. We show that the Dirac fermion DM contained in the model can explain the DAMPE excess. Meanwhile, experimental constraints from DM relic density, direct detection, LEP and anomalous magnetic moments are satisfied. 1 Introduction It is well known that new physics beyond the Standard Model (SM) is needed to accommodate two open questions: the tiny neutrino masses and the cosmological dark matter (DM) candidates. The scotogenic model, proposed by Ma [ 1,2 ], is one of the attractive candidate, which attributes the tiny neutrino masses to the radiative generation and the DM is naturally contained as intermediate messengers inside the loop. In the original models, an ad hoc Z2 or Z3 symmetry serves to guarantee the stability of DM, whereas such discrete symmetry would be broken at highscale [3]. Perhaps a more reasonable scenario is regarding the discrete symmetry as the residual symmetry originated from the breaking of a continuous U (1) symmetry at high scale. Along this line, several radiative neutrino mass models [ 4–23 ] were proposed based on gauged U (1)B−L theory, which is simplest and well-studied gauge extension of SM. Very recently, a sharp excess in the e+ +e− flux is reported by Dark Matter Particle Explorer (DAMPE), which can be interpreted as the annihilation signal of DM with the mass around 1.5 TeV [ 24 ]. For this mass scale, the DM annihilation cross section times velocity to explain the excess is in the range of 10−26–10−24 cm3/s if one further assumes the existence of a nearby dark subhalo about 0.1–0.3 kpc distant from the solar system [ 25 ]. Inspired by this assumption, various models [ 26–58 ] have been proposed. On the other hand, the associated proton–antiproton pair excess has not been observed. Therefore a leptophilic gauge theory rather than the U (1)B−L ones seems more attractive. In this work, we present a radiative neutrino mass model based on an alternative leptophilic U (1)X gauge symmetry. In the model only the right-handed SM leptons are charged under the U (1)X symmetry, resulting in the direct Yukawa couplings forbidden in the lepton sector. We will show that the Dirac neutrino masses are generated radiatively and the charged leptons, acquire masses via seesaw-like mechanism. The heavy fermions we added for anomaly-free cancellation play as the intermediate fermions in lepton mass generation. After the spontaneous symmetry breaking(SSB) of U (1)X , the dark Z2 symmetry could appear as a residual symmetry [ 59 ] such that the stability of a classes of inert particles are protected by the irrational U (1)X charge assignments from decaying into SM particles, naturally leading to stable DM candidates. The rest of this paper is organised as follows. In Sect. 2, the model is set up. In Sect. 3, we focus on DM phenomenon and its implication on DAMPE results. Conclusions are given in Sect. 4. 2 Model setup 2.1 Particle content and anomaly cancellation All the field contents and their charge assignments under SU (2)L × U (1)Y × U (1)X gauge symmetry are summarized in Table 1. First of all, the U (1)X symmetry proposed here is a right-handed leptophilic gauge symmetry since, in SM sector, only right-handed leptons carry U (1)X charges. As a result, the SM Yukawa coupling L¯ E R for charged lepton mass generation is strictly forbidden. In order to generate the minimal Dirac neutrino mass, two right-handed neutrino field νRi (i = 1, 2) are added being coupled with U (1)X , leading to the zero mass for the lightest neutrino. The direct Yukawa coupling L¯ νR is also forbidden. Instead we have introduce several Dirac fermions as the intermediated fields for lepton mass generation with their corresponding chiral components Ri/Li (i = 1–9) and FRi/Li (i = 1–4) respectively. In the scalar sector, we further add inert doublet scalars η1, η2 and one inert singlet scalar χ . An SM singlet scalar σ is added being responsible for U (1)X breaking. First and foremost, we check the anomaly cancellations for the new gauge symmetry in the model. The [SU (3)C ]2U (1)X and [SU (2)L ]2U (1)X anomalies are zero because quark and left-handed leptons are not assumed coupled to U (1)X . We then find all other anomalies are also zero because [U (1)Y ]2U (1)X : −3 × (−1)2 × 3n + 9 × (−1)2 ×2n − 9 × (−1)2 × n = 0 [U (1)X ]2U (1)Y : −3 × (3n)2 × (−1) + 9 × (2n)2 ×(−1) − 9 × n2 × (−1) = 0 [Gravity]2U (1)Y : Str [QY ]SM + (−1) − (−1) = 0 U (1)3Y : Str [Q3Y ]SM + (−1)3 − (−1)3 = 0 [Gravity]2U (1)X : −3 × 3n − 2 × 2n + 9 × 2n −9 × n + 4 × Q FL − 4 × Q FR = 0 [U (1)X ]3 : −3 × (3n)3 − 2 × (2n)3 + 9 × (2n)3 −9 × n3 + 4 × Q3FL − 4 × Q3FR = 0 (1) μ2σ + λσ vσ2 + 21 λσ vφ2 + 21 λσ η1 u Here, Q FL = (√11 + 1)n/2 and Q FR = (√11 − 1)n/2 are the U (1)X charge of FL and FR as shown in Table 1 respectively. In order to cancel anomaly, the FLi/Ri fermions acquire irrational U (1)X charge numbers. Similar scenarios also appeared in radiative inverse or linear seesaw models [ 8,9 ] where other solutions to anomaly free conditions with irrational B − L charges of mirror fermions were found. 2.2 Scalar sector The scalar potential in our model is given by V =μ2 † + μ2η1 η1†η1 + μ2η2 η2†η2 + μ2χ |χ |2 + μ2σ |σ |2 + λ ( † )2 + λη1 (η1†η1)2 + λη2 (η2†η2)2 + λχ |χ |4 + λσ |σ |4 + λη1 ( † )(η1†η1) + λη1 (η1† )( †η1) + λη2 ( † )(η2†η2) + λη2 (η2† )( †η2) + λχ |χ |2( † ) + λσ |σ |2( † ) + λη1η2 (η1†η1)(η2†η2) + λη1η2 (η1†η2)(η2†η1) + λχη1 |χ |2(η1†η1) + λσ η1 |σ |2(η1†η1) + λχη2 |χ |2(η2†η2)+λσ η2 |σ |2(η2†η2)+λχσ |χ |2|σ |2 + [μ(η1† )σ +λ(η2† )χ σ + h.c.] The scalars and σ and η1 with their vevs after SSB of U (1)X can be parameterized as G+ = vφ+φ√0φ+i Gφ , η1 = 2 σ = vσ + σ√02+ i Gσ . Then the minimum of V is determined by η+ u 1 0 , √2 + η1 + λ vφ2 + 21 λσ vσ2 + 21 u μ2η1 + λη1 u2 + 21 λσ η1 vσ2 + 21 For a large and negative μ2σ , there exists a solution with u2 vφ2 vσ2 as vσ2 , where vφ 246 GeV is the vev of the SM Higgs doublet scalar and vσ is responsible for the SSB of U (1)X symmetry. We have set a positive μ2η1 , hence the vev of η1 scalar is not directly acquired as that of and σ but induced from μ(η1† )σ term. Note that η2 and χ do not acquire vevs because of positive μ2η2 , μ2χ and the absence of linear terms, like χ σ k . The mass spectrum of scalar σ, and η1 can be obtained with the their vevs and the cross terms in Eq. (2). In the condition of u2 vφ2 vσ2 , the contributions from v and vσ to scalar masses are dominant, and mixings between η1 and other CP-even scalars are negligible small. Then the two CP-even scalars h and H with mass eigenvalues are given by m2h,H 2 2 λ vφ + λσ vσ ∓ (λ vφ − λσ vσ2 )2 + λ2σ v2 v2 , 2 φ σ with the mixing angle tan 2α = λσλvσσ2 −v λvσvφ2 , where we take scalar h as the SM-like Higgs boson and H the heavy Higgs boson. A small mixing angle sin α ∼ 0.1 is assumed to satisfy Higgs measurement [ 60 ]. Note that due to the lack of ( †η1,2)2 term, we actually have nearly degen0 erate masses for the real and imaginary part of η1,2 [ 61 ], and they are assumed to be degenerate for simplicity in the following discussion. The masses of scalar doublet η1 are μ2η1 + 21 (λφη1 + λφη1 )vφ2 + 21 λσ η1 vσ2 , 2 1 2 1 2 μη1 + 2 λφη1 vφ + 2 λσ η1 vσ . Hereafter, we take degenerate η1 scalars and m H,η1 ∼ 500 GeV for illustration. For a complete detail of mass spectrum of , σ, η1 scalars, one can refer some models which shares part of the scalar potential, e.g. Ref. [ 61 ]. On the other hand, we pay more attention to inert scalars η2 and χ which are closely related to neutrino mass generation and DM. Note that scalars η2, χ do not mix with , σ and η1. The two mass (4) (5) (6) (7) (8) (9) eigenstate of neutral complex scalars η20 and χ are obtained by S1 S2 = cos θ − sin θ sin θ cos θ λvφ vσ , sin 2θ = m2S1 − m2S2 , Mη22 + Mχ2 ± (Mη22 − Mχ )2 + λ2vφ2 vσ2 , with mass eigenvalues 1 m2S1,2 = 2 where Mη22 Mχ2 μ2η2 + 21 (λφη2 + λφη2 )vφ2 + 21 λσ η2 vσ2 , μ2χ + 21 λχφ vφ2 + 21 λχσ vσ2 . Meanwhile, the mass of inert charged scalar η2± is Mη2± 2 2 1 2 1 2 μη2 + 2 λφη2 vφ + 2 λσ η2 vσ m S1,S2,η2± ∼ 10 TeV, are assumed. As will shown in Sect. 3, the DAMPE excess favors fermion DM m F1 ∼ 1.5 TeV. Therefore, heavier inert scalars, e.g., 2.3 Lepton masses The Yukawa interactions related to charged lepton mass generation is given by L1 ⊃ y1 L¯ η1 R + y2 E¯ R L σ + y ¯ L R σ + h.c. (14) the charged lepton masses are generated though the diagram in Fig. 1. In the basis of (l¯L , ¯ L ) and (E R , R ), we obtain the 12 × 12 effective mass matrix (10) (11) (12) (13) Φ σ Then the charged lepton mass is obtained as Ml y1 y2u/(√2y). Correct charged lepton mass can be acquired e μ τ with y1,2 = 8.5 × 10−4, y1,2 = 1.2 × 10−2 and y1,2 = 5.0 × 10−2 for u = 10 GeV and y = 0.01. The Yukawa sector for Dirac neutrino mass generation is given by L2 ⊃ h1 L¯ FR η2 + h2ν¯ R FL χ † + f F¯L FR σ + h.c. (16) The effective mass matrix for active neutrinos depicted in Fig. 2 is expressed as (15) (mν )αβ = sin 2θ 8π 2 6 k=1 m2S2 − m2Fk − m2S2 log h1αk m Fk hk2β m2S2 m2Fk m2S1 m2Fk − m2S1 log m2S1 m2 k where m Fk (k = 1 − 4) denote the masses inert Dirac fermions. Typically, mν ∼ 0.1 eV can be realised with θ ∼ 10−3, h1 h2 ∼ 10−4, m F ∼ 1.5 TeV and m S1,S2 ∼ 10 TeV. From Eqs. (2), (14) and (16) , one can confirm that after the symmetry breaking with vφ and vσ there exists a residual Z2 symmetry for which the irrational U (1)X charged particles (FRi/Li , η1 and χ ) are odd while other are even. Therefore the lightest particles with irrational charges can not decay into SM particles and thus can be regarded as DM candidate. 2.4 Lepton flavor violation The new Yukawa interactions of charged lepton will induce lepton flavor violation processes at one-loop level. Taking tan 2θZ = the radiative decay α → β γ for an illustration, the corresponding branching ratio is calculated as [ 62 ] ⎨⎪⎧ 9 ⎛ ⎞ yiβ∗ yiα 1 1 F1 ⎝ h1 ∼ 10−4 with m F ∼ 1.5 TeV, mη+ ∼ 10 TeV are taken to reproduce lepton masses. Due to s2mall Yukawa coupling h and heavy mass of η2±, contribution of charged scalar η2± is suppressed. The predicted branching ratios are BR(μ → eγ ) 1.6 × 10−15, BR(τ → eγ ) 4.6 × 10−15 and BR(τ → μγ ) 9.4 × 10−13, which are clearly below current experimental limits [ 63–65 ]. 2.5 Mixing in the gauge sector (17) Since η1 is charged under both U (1)Y and U (1)X , its vev u will induce mixing between Z0 and Z0 at tree level. The resulting mass matrix in the (Z0, Z0) basis is given by [ 66 ] ⎫ 2⎬⎪ ⎪⎭ (18) (19) (20) (21) (22) (23) M 2 = 41 g2nZ2 g(vZφ2g+u2u ) 2 n2 gZ g u2 g 2n2(vσ2 + u2) . The eigenvalues of M 2 are 1 m2Z,Z = 2 with mixing angle given by 2M122 M222 − M121 . M121 + M222 ∓ (M121 − M222)2 + 4M142 , As u2 vφ2 vσ2 in this model, we have m2Z g2Z vφ2 /4, m2Z g 2n2vσ2 , and the mixing angle θZ ∼ u2/vσ2 is naturally suppressed. Typically, for u ∼ 10 GeV and vσ ∼ 10 TeV, we have θZ ∼ 10−6. Therefore, the dilepton signature pp → Z → + − at LHC is dramatically suppressed by the tiny mixing angle θZ . For light Z around EW-scale, the four lepton signature pp → + − Z → + − + − is promising at LHC [ 67 ]. As shown in next section, the DAMPE excess favors heavy Z 3 TeV. In this case, the Z can hardly be detected at LHC, but are within the reach of the 3 TeV CLIC in the e+e− → Z → μ+μ− channel [ 68 ]. 2.6 LHC signature In this subsection, we qualitatively discuss possible signatures of new particles at LHC. Since decays of η1 scalars and depend on their masses, the resulting signatures would be different. Considering the mass spectrum m < mη1 , the decay mode of η1 scalars are η10 → − +, η± → ν ±, and decay modes of are ± → ± Z , ±h, νW ±. The promising signature would be pp → + − → + − Z Z , leading to same signature as charged fermion in type-III seesaw [ 69 ]. In the opposite case m > mη1 , the decay mode of η1 scalars are η10 → + −, η1± → ±ν, and new decay modes of exotic charged fermion ± → ±η10, νη1± are also possible. Note that η1 scalars are responsible for charged fermion mass, hence η10 → τ +τ − and η1± → τ ±ν are the dominant decay mode. The promising signature would be pp → η10η10∗ → τ +τ −τ +τ −, pp → η1±η10 → τ ±ντ +τ −, similar as the lepton-specific 2HDM [ 70 ]. For the mass of scalar singlet m H ∼ 500 GeV with not too small mixing angle α ∼ 0.1, the promising signature would be gg → H → W +W −, Z Z , hh at LHC [ 71 ]. Provided m < m H,η1 , the new decay channel H → ± ∓ is also allowed. Then, the new signature gg → H → ± ∓ with ± further decaying into ± Z , νW ± is a good way to probe the corresponding Yukawa coupling y2 E¯ R L σ introduced in this model. As for the inert scalars, the most promising signature in principle would be pp → η2+η2− → + F1 + − F¯1, i.e., + − + E T , for fermion DM at LHC [ 72 ]. But actually, this dilepton signature is suppressed dramatically by heavy mass of the inert charge scalar mη± ∼ 10 TeV in our consideration [ 73 ], thus it is hard to probe2 at LHC. Similarly, the mono- j signature pp → η20η20∗ j → νν¯ F1 F¯1 j , i.e., j + E T , is also challenging at LHC. 3 DAMPE dark matter Motivated by recent DAMPE excess around 1.5 TeV, we focus on DM phenomenon in this section. Here, we consider the lightest Dirac fermion F1 as DM candidate. The relevant interactions mediated by the new gauge boson Z for DM and leptons are LZ ⊃ g Zμ Q ER E¯ R γ μ E R + QνR ν¯ R γ μνR +Q FL F¯L1γ μ FL1 + Q FR F¯R1γ μ FR1 , with mass of gauge boson Z given by m Z g nvσ . In the following numerical calculation, we will take n = 1/3 for illustration. Therefore, we have √Q ER = 1, QνR = 2/3, Q FL = (√11 + 1)/6, and Q FR = ( 11 − 1)/6. The dominant annihilation channels for DM F1 are F¯1 F1 → ¯ , ν¯ ν, Z Z . Provided m Z > m F1 , then the annihilation channel F¯1 F1 → Z Z is not allowed kinematically. Hence, F¯1 F1 → ¯ , ν¯ ν become dominant, which would be able to interpret the DAMPE e+ + e− excess when m F1 ∼ 1.5 TeV. 3.1 Constraints In this part, we summarize some relevant constraints for DAMPE DM. To research the DM phenomenon, we implement this model into FeynRules [ 74 ] package. Then, for DM relic density, we require the results calculated by micrOMEGAs4.3.5 [ 75 ] in 1σ range of Planck measurements: h2 = 0.1199 ± 0.0027 [ 76 ]. As for direct detection, the leptophilic Z will mediate DM-electron scattering at tree level, with the corresponding cross section constrained by XENON100, i.e., σe < 10−34cm2[ 77 ]. Because of XENON100 sensitive to axialvector couplings, the analytical expression for axial-vector DM-electron scattering is given by [ 78 ] σe = 3(gaF ga )2 me2 π m4 ≈ 3(gaF ga )2 Z ×3.1 × 10−39cm2, m Z 10 GeV −4 where gaF = g (Q FR − Q FL )/2 = −g /6 and ga = g Q ER /2 = g /2. For g ∼ 0.1, m Z ∼ 3 TeV, the predicted value is far below current experimental bound. Instead, we consider the loop induced DM-nucleus scattering with the cross section calculated as [ 78 ] α2 Z 2μ2N σN = 9π 3 A2m4 Z =e,μ,τ gvF gv log where μN = m N m F1 /(m N + m F1 ) is the reduced DMnucleus mass, gvF = g (Q FR + Q FL )/2 = g √11/6, gv = g Q ER /2 = g /2 and μ = m Z / gv gv is the cut-off scale. F Since current most strict direct detection constraint is performed by PandaX [ 79 ], we take Z = 54, A = 131 and (24) (25) (26) (27) Fig. 3 Left: allowed region for DAMPE DM in the g –m Z plane. The green line delimit the relic density in the 1σ range: h2 = 0.1199±0.0027, while blue and red line correspond to LEP and PandaX bound respectively. Right: predicted value of current σ v in the halo as a function of m Z . The green line satisfy the observed relic density, while the red curves are excluded by PandaX ’ g m N = 131 GeV for the target nucleus charge, mass number and mass respectively. The leptophilic Z will contribute to anomalous magnetic moments of leptons [ 80 ] (28) a g 2 m2 12π 2 m2 Z For an universal gauge–lepton coupling, the precise measurement of aμ = (27.8 ± 8.8) × 10−10 [ 81 ] set a stringent bound, i.e., g 5 × 10−3m Z /1 GeV. Meanwhile, searches for leptophilic Z at LEP in terms of fourfermion operators provide a much more stringent bound: g 2 × 10−4m Z /1 GeV [ 82 ]. 3.2 Fitting the DAMPE excess To determine the allowed parameter space under above constraints from relic density, direct detection and collider searches, we scan over the g –m Z plane while fix m F1 = 1500 GeV. The results are depicted in Fig. 3. Since the dominant annihilation channels into leptons are via s-channel, the resonance production of Z will diminish the required g coupling for correct relic density. And currently, the most stringent bound is from direct detection, which constrains Z around the resonance region. In Fig. 3, the predicted value of current σ v in the halo is also shown. Slightly below the resonance, the Breit–Wigner mechanism [ 83 ] greatly enhances the annihilation cross section. In contrast, we see a strong dip 10 23 10 24 10 25 3 s just above the resonance. Considering the fact that DAMPE excess favor σ v > 10−26cm3/s as well as PandaX has excluded the region m Z < 2810 GeV ∪ m Z > 3380 GeV, the possible region to interpret DAMPE excess falls in the range m Z ∈ [2810, 3000] GeV. Based on the above analysis, we select a benchmark point (see Table 2) to fit the sharp DAMPE excess by taking into account contributions from both nearby subhalo and Galactic halo. In our numerical calculation, we respectively use GALPROP [87,88] and micrOMEGAs packages [ 75 ] to evaluate the background flux coming from various astrophysical sources and the flux due to DM annihilation in Galactic halo. While for subhalo contribution, we numerically solve following steady-state diffusion equation [ 84 ] ∂ −∇ · K (E )∇ f (x , E ) − ∂ E ! b(E ) f (x , E ) = Q(x , E ) , with the source term " σ v Q(x , E ) = 2m2DM ρ2(r )d V δ3(x − xsub) , by using Green function method. In Eqs. (29) and (30), K (E ) = K0(E /E0)δ is the diffusion coefficient, b(E ) = E 2/(E0τE ) is the positron loss rate due to the synchrotron radiation and inverse Compton scattering, σ v the thermal averaged cross section at present, ρ(r ) and xsub the density profile and location of nearby subhalo, respectively. Here we (29) (30) adopt propagation parameters as [ 25 ]: K0 = 0.1093 kpc2 Myr−1, δ = 1/3, L = 4 kpc (the half height of the Galactic diffusion cylinder), τE = 1016 s (the typical loss time) and E0 = 1 GeV. In addition, we assume both Galactic halo and subhalo are follow NEW density profile [ 85,86 ]: ρs ρ(r ) = (r/rs )(1 + r/rs )2 . The Galactic halo is normalized by the local density ρ at Sun orbit R , which are respectively fixed as ρ = 0.4 GeVcm−3 and R = 8.5 kpc. While for nearby subhalo, the parameters ρs and rs can be determined by its viral mass Mvir. The fitting result for our benchmark point is presented in Fig. 4 together with DAMPE data points. From which, we find that a nearby subhalo with a distance of 0.1 (0.3) kpc and the viral mass 3 × 107 M (3 × 108 M ) can account for the DAMPE excess for our model. (31) In this paper, we propose an anomaly-free radiative seesaw model with an alternative leptophilic U (1)X gauge symmetry. Under the U (1)X symmetry, only right-handed leptons are charged. Charged leptons acquire mass via the type-I seesaw-like mechanism with heavy intermediate fermions added also for anomaly-free cancellation. Meanwhile, tiny neutrino masses are generated at one-loop level with DM candidate in the loop. Provided all other particles are heavy enough, the dominant annihilation channel for DM F1 is F¯1 F1 → ¯ , ν¯ ν mediated by the new leptophilic gauge boson Z . Motivated by the observed DAMPE e+ + e− excess around 1.5 TeV, we fix m F1 = 1.5 TeV while consider possible constraints from relic density, direct detection and collider searches. Under all these constraints, a benchmark points, i.e., m Z = 2950 GeV, is chosen from the viable region m Z ∈ [2810, 3000] GeV. After fitting to the observed spectrum, we find that the DAMPE excess can be explained by a nearby subhalo with a distance of 0.1 (0.3) kpc and the viral mass 3 × 107 M (3 × 108 M ). Acknowledgements The work of Weijian Wang is supported by National Natural Science Foundation of China under Grant number 11505062, Special Fund of Theoretical Physics under Grant number 11447117 and Fundamental Research Funds for the Central Universities under Grant number 2014ZD42. We thank Qiang Yuan for help on DAMPE spectrum fitting. 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Zhi-Long Han, Weijian Wang, Ran Ding. Radiative seesaw model and DAMPE excess from leptophilic gauge symmetry, The European Physical Journal C, 2018, 216, DOI: 10.1140/epjc/s10052-018-5714-3