Radiative Seesaw Model and DAMPE Excess from Leptophilic Gauge Symmetry

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.


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 Z 2 or Z 3 symmetry serves to guarantee the stability of DM, whereas such discrete symmetry would be broken at higha e-mail: sps_hanzl@ujn.edu.cn b e-mail: wjnwang96@aliyun.com c e-mail: dingran@mail.nankai.edu.cn scale [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][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][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 cm 3 /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  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 Z 2 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 Table 1 Contents of relevant particle fields. We have set i = 1, 2; m = 1 − 9 and α = 1 − 4 to satisfy the anomaly free condition Lepton fields Scalar fields Ri/Li (i = 1-9) and F Ri/Li (i = 1-4) respectively. In the scalar sector, we further add inert doublet scalars η 1 , η 2 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 ] 2 U (1) X and [SU (2) L ] 2 U (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 Here, Q F L = ( √ 11 + 1)n/2 and Q F R = ( √ 11 − 1)n/2 are the U (1) X charge of F L and F R as shown in Table 1 respectively. In order to cancel anomaly, the F Li/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.

Scalar sector
The scalar potential in our model is given by The scalars and σ and η 1 with their vevs after SSB of U (1) X can be parameterized as Then the minimum of V is determined by For a large and negative μ 2 σ , there exists a solution with where v φ 246 GeV is the vev of the SM Higgs doublet scalar and v σ is responsible for the SSB of U (1) X symmetry. We have set a positive μ 2 η 1 , hence the vev of η 1 scalar is not directly acquired as that of and σ but induced from μ(η † 1 )σ term. Note that η 2 and χ do not acquire vevs because of positive μ 2 η 2 , μ 2 χ and the absence of linear terms, like χσ k .
The mass spectrum of scalar σ, and η 1 can be obtained with the their vevs and the cross terms in Eq. (2). In the condition of u 2 v 2 φ v 2 σ , the contributions from v and v σ to scalar masses are dominant, and mixings between η 1 and other CP-even scalars are negligible small. Then the two CP-even scalars h and H with mass eigenvalues are given by with the mixing angle 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 ( † η 1,2 ) 2 term, we actually have nearly degenerate masses for the real and imaginary part of η 0 1,2 [61], and they are assumed to be degenerate for simplicity in the following discussion. The masses of scalar doublet η 1 are Hereafter, we take degenerate η 1 scalars and m H,η 1 ∼ 500 GeV for illustration. For a complete detail of mass spectrum of , σ, η 1 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 η 2 and χ which are closely related to neutrino mass generation and DM. Note that scalars η 2 , χ do not mix with , σ and η 1 . The two mass Fig. 1 Charged lepton mass generation eigenstate of neutral complex scalars η 0 2 and χ are obtained by with mass eigenvalues where Meanwhile, the mass of inert charged scalar η ± 2 is As will shown in Sect. 3, the DAMPE excess favors fermion DM m F 1 ∼ 1.5 TeV. Therefore, heavier inert scalars, e.g., m S 1 ,S 2 ,η ± 2 ∼ 10 TeV, are assumed.

Lepton masses
The Yukawa interactions related to charged lepton mass generation is given by 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 Then the charged lepton mass is obtained as M l y 1 y 2 u/( √ 2y). Correct charged lepton mass can be acquired with y e 1,2 = 8.5 × 10 −4 , y μ 1,2 = 1.2 × 10 −2 and y τ 1,2 = 5.0 × 10 −2 for u = 10 GeV and y = 0.01.
The Yukawa sector for Dirac neutrino mass generation is given by The effective mass matrix for active neutrinos depicted in Fig. 2 is expressed as where m F k (k = 1 − 4) denote the masses inert Dirac fermions. Typically, m ν ∼ 0.1 eV can be realised with θ ∼ 10 −3 , h 1 h 2 ∼ 10 −4 , m F ∼ 1.5 TeV and m S 1 ,S 2 ∼ 10 TeV. From Eqs. (2), (14) and (16) , one can confirm that after the symmetry breaking with v φ and v σ there exists a residual Z 2 symmetry for which the irrational U (1) X charged particles (F Ri/Li , η 1 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.

Lepton flavor violation
The new Yukawa interactions of charged lepton will induce lepton flavor violation processes at one-loop level. Taking the radiative decay α → β γ for an illustration, the corresponding branching ratio is calculated as [62] where C h = sin α and C H = cos α. And the loop functions F 1,2 (x) are given by According to previous discussion, y e 1,2 = 8.5 × 10 −4 , y

Mixing in the gauge sector
Since η 1 is charged under both U (1) Y and U (1) X , its vev u will induce mixing between Z 0 and Z 0 at tree level. The resulting mass matrix in the (Z 0 , Z 0 ) basis is given by [66] The eigenvalues of M 2 are with mixing angle given by As σ , and the mixing angle θ Z ∼ u 2 /v 2 σ is naturally suppressed. Typically, for u ∼ 10 GeV and v σ ∼ 10 TeV, we have θ Z ∼ 10 −6 . Therefore, the dilepton signature pp → Z → + − at LHC is dramatically suppressed by the tiny mixing angle θ Z . 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].
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,η 1 , 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 y 2Ē R L σ introduced in this model. As for the inert scalars, the most promising signature in principle would be pp → η + 2 η − 2 → + F 1 + −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 m η ± 2 ∼ 10 TeV in our consideration [73], thus it is hard to probe at LHC. Similarly, the monoj signature pp → η 0 2 η 0 * 2 j → νν F 1F1 j, i.e., j + E T , is also challenging at LHC.

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 F 1 as DM candidate. The relevant interactions mediated by the new gauge boson Z for DM and leptons are with mass of gauge boson Z given by m Z g nv σ . In the following numerical calculation, we will take n = 1/3 for illustration. Therefore, we have Q E R = 1, Q ν R = 2/3, Q F L = ( √ 11 + 1)/6, and Q F R = ( √ 11 − 1)/6. The dominant annihilation channels for DM F 1 arē Provided m Z > m F 1 , then the annihilation channelF 1 F 1 → Z Z is not allowed kinematically. Hence,F 1 F 1 →¯ ,νν become dominant, which would be able to interpret the DAMPE e + + e − excess when m F 1 ∼ 1.5 TeV.

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: h 2 = 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., σ e < 10 −34 cm 2 [77]. Because of XENON100 sensitive to axialvector couplings, the analytical expression for axial-vector DM-electron scattering is given by [78] where g a F = g (Q F R − Q F L )/2 = −g /6 and g a = g Q E R /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] where μ N = m N m F 1 /(m N + m F 1 ) is the reduced DMnucleus mass, g v F = g (Q F R + Q F L )/2 = g √ 11/6, g v = g Q E R /2 = g /2 and μ = m Z / g v F g v is the cut-off scale. Since current most strict direct detection constraint is performed by PandaX [79], we take Z = 54, A = 131 and

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 F 1 = 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 just above the resonance. Considering the fact that DAMPE excess favor σ v > 10 −26 cm 3 /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] with the source term by using Green function method. In Eqs. (29) and (30), 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 x sub the density profile and location of nearby subhalo, respectively. Here we  [24]. Corresponding total fluxes (background + Galactic halo + subhalo) are also shown by solid lines with the same colors. Here we take solar modulation potential as = 700 MV for illustration. For comparison, the direct measurements from AMS-02 [89] and Fermi-LAT [90] experiments, as well as the indirect measurement by [91,92] are also shown. The error bars of DAMPE, AMS-02 and Fermi-LAT include both systematic and statistical uncertainties adopt propagation parameters as [25]: K 0 = 0.1093 kpc 2 Myr −1 , δ = 1/3, L = 4 kpc (the half height of the Galactic diffusion cylinder), τ E = 10 16 s (the typical loss time) and E 0 = 1 GeV. In addition, we assume both Galactic halo and subhalo are follow NEW density profile [85,86]: 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 ρ s and r s can be determined by its viral mass M vir . 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 × 10 7 M (3 × 10 8 M ) can account for the DAMPE excess for our model.

Conclusion
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 F 1 isF 1 F 1 →¯ ,νν mediated by the new leptophilic gauge boson Z . Motivated by the observed DAMPE e + +e − excess around 1.5 TeV, we fix m F 1 = 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 × 10 7 M (3 × 10 8 M ).