$Z'$ Portal Dark Matter in $B-L$ Scotogenic Dirac Model

In this paper, we perform a detail analysis on the phenomenology of $Z'$ portal scalar and Dirac fermion dark matter in $B-L$ scotogenic Dirac model. Unconventional $B-L$ charge $Q$ is assigned to the right-handed neutrino $\nu_R$ in order to realise scotogenic Dirac neutrino mass at one-loop level, where three typical value $Q=-\frac{1}{4},-4,\frac{3}{2}$ are chosen to illustrate. Observational properties involving dilepton signature at LHC, relativistic degrees of freedom $N_\text{eff}$, dark matter relic density, direct and indirect detections are comprehensively studied. Combined results of these observables for the benchmark scenarios imply that the resonance region $M_\text{DM}\sim M_{Z'}/2$ is the viable parameter space. Focusing on the resonance region, a scanning for TeV-scale dark matter is also performed to obtain current allowed and future prospective parameter space.


I. INTRODUCTION
Despite the unambiguous confirmation of non-zero neutrino masses and mixings, the nature of neutrinos, which could be Majorana or Dirac, is still an open question.It is usually argued that tiny neutrino masses may naturally originate from the effective ∆L = 2 lepton number violating (LNV) operators [1], so the Majorana scenario seems more promising.However, the neutrinoless double beta decay (0ν2β) experiments, aimed to search for ∆L = 2 LNV signatures, have been performed without any positive result so far and the possibility that neutrinos are Dirac particles can not be excluded.In the standard model (SM) along with three copies of ν R , Dirac neutrinos can acquire their masses via direct Yukawa coupling y ν LHν R .In order to generate sub-eV neutrino masses, the coupling constants y ν have to be fine tuned to 10 −12 order.Consequently, if neutrinos are Dirac particles, certain new physics beyond SM should exist to explain the extremely small Yukawa coupling constants in a natural way.Motivated by these considerations, interest in Dirac neutrino masses has been revived recently and several models [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] are proposed at tree or loop level.
On the other hand, the connections between LNV effects and the nature of neutrinos are often overlooked.For example, Dirac neutrinos are compatible with standard baryogenesis processes [20,21], where the non-perturbative effect breaks the baryon (B) and lepton (L) number each by three units while B − L number remains conserved.As is well known, the simplest anomaly-free B − L extension of SM includes three copies of ν R having −1 charge under U (1) B−L symmetry.Then the Majorana neutrino will not arise if U (1) B−L symmetry never breaks down by two units, i.e., ∆(B − L) = 2, which explains why neutrinos are Dirac fermions.In Ref. [22,23], the quartic LNV operators i.e., ∆(B −L) = 4 processes are considered as new sources for leptogenesis even if neutrinos are Dirac particles.
However, the simple B−L extension of SM model can not explain the extremely small Yukawa coupling constant in the presence of y ν LHν R term.It is reasonable to ask if gauged U (1) B−L symmetry can play a key role in the generation of naturally small Dirac neutrino masses.Recently, a class of B − L scotogenic models are proposed by us in Ref. [10] where the Dirac neutrino masses are generated at loop level.The Greek "scotogenic" means darkness.The original model, firstly propose by Ma [24], is to accommodate two important missing pieces of SM: no-zero Majorana neutrino mass and dark matter (DM) in a unified framework.The main idea is based on the assumption that the DM candidates, whose the stability is protected by an ad hoc Z 2 or Z 3 [25] symmetry, can serve as intermediate messengers propagating inside the loop diagram in neutrino mass generation.In the scotogenic Dirac model, the B −L quantum number of ν R are appropriately assigned so that the y ν LHν R interaction and Majorana term ν R ν c R are forbidden.Some new Dirac fermions are introduced to be intermediate particles carrying B − L quantum numbers.Since all new particles are SM singlets, we only need to consider the [U (1) B−L ] × [Gravity] 2 and [U (1) B−L ] 3 anomaly free conditions.Then the effective ν L ν R operator is induced after the spontaneous breaking of U (1) B−L at high scale.Moreover, the discrete Z 2 or Z 3 symmetry could appear as a remnant symmetry of gauged U (1) B−L symmetry, leading to DM candidates.
In previous studies [10], we focus on scalar interactions of DM in B − L scotogenic Dirac model.As complementary, this paper will concentrate on gauge interactions of DM.Early researches on various Z portal DM can be found in Refs.[26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41].As for Majorana fermion DM in conventional B − L scotogenic Majorana model for neutrino masses [42], no observable signatures are expected at incoming direct or indirect detection experiments except for LHC dilepton signature of Z .In contrast, the Z portal scalar DM in B − L scotogenic Dirac model could be tested at direct detection experiments, and Dirac fermion DM could be further probed by indirect detection.Therefore, if the ongoing DM experiments observe some positive signatures, the nature of DM might be revealed.
Moreover, the existence of right-handed neutrino ν R will also contribute to effective number of relativistic degrees of freedom N eff for neutrinos via Z portal, which can also be used to verify this model.
Note that although the well studied simplified Z portal DM approach is constructive, there might be some different predictions for a complete model, especially when a new observable as N eff in this model is involved.So in this paper, we perform a comprehensive analysis on the phenomenology of Z portal dark matter in B − L scotogenic Dirac model in aspect of collider signatures, dark radiation, relic density, direct and indirect detection.
The rest of the paper is organised as following.In Sec II, we briefly review the minimal gauged U (1) B−L scotogenic Dirac model.Collider signatures from LHC are discussed in Sec III.The exclusion limits are derived from the latest dilepton signature search.Dark radiation observable N eff and corresponding limits are then considered in Sec.IV.The DM phenomenology in aspect of relic density, direct detection and indirect detection are detail studied in Sec.V for some benchmark scenarios.In Sec.VI, we first present the combined results for certain benchmark scenarios.And then based on the combined results, we perform a scan over the resonance region M DM M Z /2.Finally, the conclusion is devoted to Sec.VII.

II. THE MODEL
The gauged U (1) B−L scotogenic Dirac neutrino mass models are systematically studied in Ref. [10].In this paper, we concentrate on the minimal U (1) B−L scotogenic Dirac model, i.e., model A 1 in Ref. [10].
The relevant particle contents and corresponding charge assignment are explicitly shown in Table I.In addition to three generations of right-handed neutrino ν R with B − L charge Q, this model further employs Obviously, the anomalies has been canceled for per generation fermions, since F L,R has opposite B − L charge comparing with ν L,R .Therefore, Q can be regarded as a free parameters apart from the following exceptions: • To forbid the SM direct Yukawa coupling term νL ν R φ 0 , the condition Q = −1 should be imposed.
• To forbid Majorana mass terms , the conditions Q = 0, −1/3 and 1 should also be satisfied respectively.
• Similarly, forbidding the Once an appropriate Q is assigned, the residual Z 2 symmetry appears in Eq. 3, 4 and 6, under which the parity is odd for inert particles (η, χ, F L/R ) and even for all other particles.In the following, we consider , −4 and 3 2 for illustration.The relevant new gauge interactions are dictated by One-loop generation of Dirac neutrino mass. where The complete scalar potential is given by [10] where µ 2 X (X = Φ, η, χ, σ) are all taken to be positive, and µ is also positive by proper re-phasing η and χ.Due to the lack of (Φ † η) 2 term, masses of η 0 R and η 0 I are degenerate as a complex scalar η 0 .Because η 0 has direct coupling with Z, only the singlet χ is suitable for scalar DM.To concentrate on Z -portal DM in this paper, the Higgs-portal λ Φχ , λ Φσ and λ χσ terms are assumed to be tiny.
After spontaneous symmetry breaking, the scalar Φ and σ are denoted as in unitarity gauge.Here, v φ = 246 GeV is the electroweak scale.Then the λ Φσ term induces mixing between φ 0 and ϕ 0 with mixing angle α, resulting to mass eigenstates h 0 and H 0 [44].In this paper, we regard h 0 as the 125 GeV Higgs boson discovered at LHC [45,46], and H 0 is a heavier scalar singlet [47].
Meanwhile, the Z 2 -odd scalar χ and η 0 mix into H 0 1 and H 0 2 with mixing angle β.In this paper, we assume M H 0 1 < M H 0 2 and β 1, so H 0 1 is dominant by χ-component and is regarded as scalar DM candidate.Besides, the Z 2 -odd charged scalar H ± 2 (= η ± ) do not mix with other particles.The Yukawa interactions accounting for radiative Dirac neutrino mass are given as [10]  Hence, VEV of σ will induce Dirac fermion mass M F = y F v σ / √ 2, the lightest of which, i.e., F 1 serves as fermion DM candidate.As shown in Fig. 1, the above Yukawa interactions will induce Dirac neutrino mass at one-loop level.The neutrino mass matrix can be expressed as To accommodate neutrino mass around 0.1 eV, β ∼ 10 −4 , y 1,2 ∼ 0.01 with masses of Z 2 -odd particles at TeV-scale are required.
In the following, we concentrate on the phenomenology of Z portal DM.In this case, there are only four The precise measurement of four-fermion interactions at LEP requires [48] M Z g 7 TeV.
In the limit that masses of SM fermions f (f ≡ q, l, ν) are small compared with the Z mass, the decay width of Z into fermion pair f f is given by where N f C is the number of colours of the fermion f , i.e., N l,ν C = 1, N q C = 3.Then the branch ratios of Z decay into each final states take the ratios as The Q-value has a great impact on the decay properties of Z .According to Eq. 10, the dominant decay A viable pathway to verify the Q-value is The dashed line is the upper limit from Ref. [52].
via the measurement of the ratio BR(Z In Table II, we depict the explicit value of branching ratios of Z for Q = − 1 4 , −4 and 3 2 respectively.For Q = − 1 4 , the dominant decay mode Z → + − has a branching ratio of 0.45.In comparison, the branching ratio of this dilepton mode diminishes to 0.10 when Q = −4, and the invisible decay mode Z → ν ν with a branching ratio 0.83 becomes the dominant one. The most promising signature of U (1) B−L gauge boson at LHC is the dilepton signature pp → Z → + − ( = e, µ) [49].Searches for the new gauge boson Z in the dilepton final states have been performed by ATLAS [50] and CMS [51].Due to no observation of any excess, the most stringent upper limit on the cross-section times branching ratio (σBR) has been set by ATLAS using 36.1 fb −1 data at √ s = 13 TeV LHC [52].Here, this upper limit will be interpreted in the the frame work of B − L scotogenic Dirac model.
In carrying out the theoretical cross section of the dilepton signature, we implement the scotogenic Dirac model into FeynRules [57] and employ MadGraph5 aMC@NLO [58] with NNPDF By comparing the theoretical prediction and experimental limit, one can easily obtain the exclusion limit in the g − M Z plane, which is depicted in Fig. 3 for 2 in the U (1) B−L scotogenic Dirac model.The dashed line corresponds to the LEP limit in Eq. 8.
exclusion limit for Q = −4 are the weakest, while exclusion limits are similar for 4 for an instance, the lower limit of Z mass is about 4 TeV with O(1) g -value, and this lower limit could down to about 3 TeV with g ∼ 0.1.In the lower mass region M Z 3 TeV, the limit from LHC dilepton signature is more stringent than LEP.But for M Z 4 TeV, the LEP limit becomes severer than the LHC limit.

IV. DARK RADIATION
In this B − L scotogenic Dirac model, the light right-handed neutrinos ν R serve as a new form of dark radiation in the expansion of the universe, which will affect the effective number of relativistic degrees of freedom N eff for neutrinos.Here, we use the combined Planck TT+lowP+BAO data, N eff = 3.15 ± 0.23 [60].One naively may worry that the existence of three generation of ν R is in conflict with observed data, since the relativistic degrees of freedom for neutrinos are doubled.Actually, due to different gauge interactions, ν R could decouple earlier than ν L , leading to a suppression of contribution to N eff from ν R .
The corresponding theoretical contribution of ν R to ∆N eff = N eff − N SM eff is given by [61,62] where N SM eff = 3.046 comes from the contribution of SM left-handed neutrinos [63], and g(T ) is the effective number of degrees of freedom at temperature T [64].The decoupling temperature of left-handed The shaded pink region is allowed by current experiments [60].
corresponds to three ν L , e ± and photon [65].In the gauged U (1) B−L scotogenic Dirac model, ν R are in equilibrium with SM fermions via the new gauge boson Z , therefore the interaction rate of ν R is calculated as [64] where s = 2pq(1 − cos θ), v = (1 − cos θ) with θ the relative angle of the colliding right-handed neutrinos.
and the number density of the right-handed neutrinos n ν R with spin number g ν R = 2 is given by In the limit M 2 Z s, the annihilation cross section in Eq. 12 is given by Including the contribution of three right-handed neutrinos, the Hubble parameter is now derived by The dashed line corresponds to the LEP limit in Eq. 8.
The right-handed neutrinos decouple when . Solving this condition numerically, one can obtain T ν R dec , and then ∆N eff via Eq.11.The results of ∆N eff as a function of M Z are exhibited in Fig. 4. According to Eq. 14, for a larger g , |Q| or a smaller M Z , the corresponding annihilation cross section is larger, resulting in a smaller decoupling temperature T ν R dec [64].Therefore, g(T ν R dec ) is smaller and then ∆N eff is larger.Such changes are clearly shown in Fig. 4. We notice that for TeV-scale Z , the minimum value of ∆N eff is about 0.2, corresponding to a plateau around g(T ν R dec ) ∼ 86 when 3 GeV T ν R dec 10 GeV, and are hopefully within the reach of future Euclid experiment [66].
By requiring ∆N eff = 0.564, one can further derive the exclusion limit in the g − M Z plane, which is shown in Fig. 5. Depending on |Q|, the exclusion limits are approximately M Z /g 3.5, 9.5, 21 TeV for , −4 respectively.In contrast to LHC limits, a larger |Q| will lead to a more stringent limit from ∆N eff .And for Q = 3 2 and −4, the exclusion limits have already exceeded LEP limit.

V. DARK MATTER
In this section, we investigate the phenomenology of Z -portal DM, which is determined by relevant interactions presented in Eq. 3. As stated in Sec.II, we consider H 0 1 and F 1 as scalar and fermion DM candidate respectively.DM observables, such as the DM relic density, the DM-nucleon cross section and the thermally averaged annihilation cross section σv , are calculated with the help of micrOMEGAs [67] for a more precise results.In order to illustrate the dependence of DM observables on certain variables, we first carry out the results of some benchmark scenarios in this section.The combined analysis of different observables and a scan over corresponding parameter space will be performed in Sec.VI.

A. Relic Density
Here, we consider the conventional WIMP DM freeze-out scenario.As long as the temperature T is high enough, the DM candidate is in thermal equilibrium with the primordial thermal bath via the new gauge interaction.And after the DM interaction rate is smaller than the expansion rate of the Universe, it is decoupled.The DM number density n is then calculated by solving the following Boltzmann equation [68] dn dt where H is the Hubble parameter, and n eq is the thermal equilibrium value.In the following numerical results, we use micrOMEGAs [67] to solve the above Boltzmann equation and determine the DM relic density.Approximately, one can estimate the DM relic density by which will be taken for qualitative illustration, since it is more intuitional.Moreover, the DM relic density measured by Planck is Ωh 2 = 0.1199 ± 0.0027 [60].
In the B − L scotogenic Dirac model, the DM candidate H 0 1 or F 1 can annihilate into Z 2 -even fermion pairs via the exchange of Z in the s-channel.Neglecting masses of the final state fermions, the corresponding thermal averaged annihilation cross sections are given by [69] σv for scalar DM H 0 1 and fermion DM F 1 respectively.Here, , can be extracted from Eq. 3. In the mean time, since the scalar DM H 0 1 is dominant by the χ-component, we set Clearly, the annihilation cross sections for scalar DM is velocity suppressed.If DM is heavier than Z , then DM pair can also annihilate into Z pair, TeV.The cyan band corresponds to Planck measured result Ωh 2 = 0.1199 ± 0.0027 [60].leading to annihilation cross sections as [69] σv Therefore, free parameters involved in DM relic density are gauge coupling g , B − L charge Q, gauge boson mass M Z , and DM mass M H 0 1 ,F 1 , impacts of which are discussed in the following.First, we consider the relic density of scalar DM H 0 1 , which is shown in Fig. 6.Because of resonant production of Z in the s-channel, all the sub-figures exhibit sharp dips around M H 0 1 ∼ M Z /2 in the relic density curves, which is a clear behaviour of Eq. 18.And the resonance dips are broadened with increasing gauge g , absolute value of H 0 1 charge |Q H 0 1 | = |Q − 1| and gauge boson mass M Z .In addition, around M H 0 1 ∼ M Z , one also sees a decrease of relic density as increase of DM mass.Because, when M H 0 1 > M Z , the velocity independent annihilation process H 0 1 H 0 * 1 → Z Z (see Eq. 20) is kinematically allowed and dominates over the velocity dependent process H 0 1 H 0 * 1 → Z * → f f (see Eq. 18).More specifically, in Fig. 6 (a), we fix Q = − 1  4 , M Z = 2 TeV, and vary g = 0.1, 0.2, 0.5.It is obvious that a larger g leads to a larger annihilation cross section, thus a smaller relic density.By requiring TeV.The cyan band corresponds to Planck measured result Ωh 2 = 0.1199 ± 0.0027 [60].
Ωh 2 0.12 to satisfy observed value, one actually finds two viable DM mass M H 0 1 , one below M Z /2 and the other above M Z /2.In Fig. 6 (b), g = 0.2, M Z = 2 TeV are fixed and Q = − 1 4 , −4, 3 2 is varied.From Eq. 18 and Eq. 20, one is aware that the annihilation cross sections are positively correlated with the cross section of annihilation process H 0 1 H 0 * 1 → Z Z is sufficient large to reach Ωh 2 0.12.Fig. 6 (c) depicts the impact of gauge boson mass M Z when fixing Q = − 1 4 and g = 0.2.Although the resonance dips exist for all kinds of M Z , the minimum value of relic density increases as M Z increases.And eventually for M Z = 6 TeV, the minimum value of relic density is larger than the observed value ∼ 0.12.
Then, we move on to fermion DM.The results are shown in Fig. 7. Similar to scalar DM, resonance dips appear around M F 1 ∼ M Z /2.But comparing with scalar DM, the relic density of fermion DM is easier to reach Ωh 2 ∼ 0.12 with same values of free parameters, because the s-channel process is not velocity suppressed (see Eq. 18 and Eq.19).In the limit of M 2 So with increasing M F 1 , the process F 1 F1 → Z Z gradually becomes more important.Besides, the annihilation cross section, thus the resulting relic density, tends to a constant when M F 1 increases.
The impacts of gauge coupling g , ν R 's B − L charge Q and gauge boson mass M Z are shown in Fig. 7 (a), (b) and (c) respectively.Again, effects of changing g and M Z are similar as scalar DM, but not Q.
When M F 1 < M Z , Eq. 19 indicates that the vector coupling V FIG. 8. Spin-independent cross section of scalar DM H 0 1 scattering on nucleon.Left: as a function of Here, g is determined by yielding correct relic density Ωh 2 0.12.Experimental bounds come from XENON1T2017(green) [71], PandaX2017(pink) [72], and XENON1T(2t•y)(brown) [73].gion can escape current PandaX2017 limit [72], it is within the reach of future XENON1T(2•y) [73].For , it can even avoid future XENON1T(2•y) limit [73] in the resonance region.Right panel of Fig. 8 depicts σ SI as a function of M H 0 1 , while we fix Q = − 1 4 and vary M Z = 2, 4, 6 TeV.Basically, the larger M Z is, the easier it will be to escape direct detection bounds, due to the bounds is less stringent with increasing DM mass.For M Z = 4, 6 TeV, except for the region quite close to the resonance region M H 0 1 M Z /2, XENON1T(2•y) [73] could probe most of the parameter space.
Then, we come to fermion DM F 1 .The results are shown in Fig. 9. Similar as scalar DM, only the resonance region can escape direct detection limits.But in this region, the resulting σ SI is much smaller, so that both varying Q = − 1 4 , −4, 3 2 and varying M Z = 2, 4, 6 TeV could easily avoid current and even future limits.Since only the vector coupling V F 1 = (1 − Q)/2 is involved in relic density (see Eq. 18) and spin-independent DM-nucleon scattering (see Eq. 23) ) corresponds to the smallest B −L charge.Hence, the largest g is required to reproduce correct relic density, leading to largest scattering cross section.
In summary, only the resonance region M DM ∼ M Z /2 can satisfy direct detection limits by requiring the free parameter set yielding correct relic density for both scalar and fermion DM.Moreover, in the resonance region, we find that a larger effective absolute value of B − L charge, i.e., |Q FIG. 9. Same as Fig. 8, but for fermion DM F 1 .
cross section.And comparing with scalar DM, fermion DM is easier to escape direct detection limits.

C. Indirect Detection
According to the results of direct detection in Sec.V B, the so-called secluded DM scenario M DM > M Z [75,76] has already been excluded.So in the B − L scotogenic Dirac model, DM pair can only annihilate into SM fermion pair through the gauge boson Z in the s-channel nowadays.Then, this leads to high energy gamma-rays, which is detectable at Fermi-LAT [77], H.E.S.S. [78,79], and the forthcoming CTA [80,81].Provided DM 100% annihilating into b b or τ + τ − with an annihilation cross section of 3×10 −26 cm 3 s −1 , Fermi-LAT has excluded DM mass below 100 GeV [77].As for TeV-scale DM, H.E.S.S.
has excluded 500 GeV M DM 2 TeV when DM solely annihilates into τ + τ − final states also assuming an annihilation cross section of 3 × 10 −26 cm 3 s −1 [78,79].To illustrate prospect of indirect detection, we also take into account the upcoming CTA experiment [80] and considering the most optimistic limits in Ref. [81], which in principle could probe M DM 10 TeV for a 100% branching ratio into τ + τ − and an annihilation cross section of 3 × 10 −26 cm 3 s −1 .Since for TeV-scale DM, the τ + τ − channel provides the most stringent limit, we will take this channel to illustrate for simplicity.A more appropriate pathway is taking into all annihilation channels [30,87].
where BR f is the branching ratio of Z into fermion pair f , dN f γ /dE γ is the differential γ-spectrum from fermion pair f , J ann is the astrophysical J-factor.And the annihilation cross sections are presented in Eq. 18 for scalar DM and Eq.19 for fermion DM, respectively.Because the annihilation cross section today is velocity suppressed for scalar DM, the corresponding indirect detection signatures are usually neglected [82].As for fermion DM, the annihilation cross section is not suppressed, hence indirect detection signatures are in principle promising.Furthermore, since only the resonance region M DM ∼ M Z /2 is viable under constraints from relic density and direct detection, the naive estimation of DM annihilation cross section with Eq. 17, i.e., σv 3 × 10 −26 cm 3 s −1 , is invalid [83,84].In this case, the annihilation cross section could be enhanced via the Breit-Wigner mechanisms [85,86], so as a consequence the indirect detection limits are strengthened.
We start with scalar DM as well.The annihilation cross sections for the τ + τ − channel are shown in , −4 are enhanced to the order of 10 −24 cm 3 s −1 , which has already exceed Fermi-LAT limit.The maximum value for Q = 3  2 is one order of magnitude smaller.Although this is lower than Fermi-LAT limit, it is still larger than H.E.S.S. limit.Future CTA limit will push the limit down to about 5 × 10 −27 cm 3 s −1 , hence large part of the resonance regions are detectable.Moreover, when M F 1 is slightly lighter than M Z /2, the cross sections dramatically diminish to 3 × 10 −29 cm 3 s −1 , 8 × 10 −30 cm 3 s −1 ,6 × 10 −31 cm 3 s −1 for Q = 3  2 , − 1 4 , −4 respectively.Therefore, this region can easily escape indirect detection limit.Right panel of Fig. 11 shows σv τ + τ − as a function of M H 0 1 , while we fix Q = − 1 4 and vary M Z = 2, 4, 6 TeV.One can see that only M Z = 2 TeV could exceed Fermi-LAT limit, since as M H 0 1 (M Z ) getting bigger, the Fermi-LAT limit is less stringent while the enhancement effect is weaker.With clearly enhanced cross section, the resonance regions are all within the reach of H.E.S.S. and CTA.
With fixed DM mass and B − L charge Q, the green lines further satisfy relic density condition Ωh 2 0.12 [60].To make sure the model is perturbative, we also require |g detection, now the resonance region can escape PandaX2017 limit, but most region are still in the reach of XENON1T(2t•y).
Then, the combined results for fermion DM F 1 is shown in  12, but for fermion DM F 1 .In addition, the pink and brown curves are excluded by H.E.S.S. [78] and future CTA [81] indirect detection experiments.
which can be easily obtained from Fig. 11 (a).The benchmark scenario is shown in Fig. 13 (a).It is amazing that the combined result of LHC@13TeV and H.E.S.S. has already excluded M F 1 M Z /2.So neither direct detection nor indirect detection experiment will observe positive signature.And the only way to probe the allowed region M F 1 M Z /2 is via dilepton signature at ongoing LHC.Similar arguments are also true for benchmark scenario M F 1 = 1 TeV with Q = 3 2 as shown in Fig. 13 (c).Note in this scenario, the constraint from ∆N eff is more stringent than current direct detection three generations of Dirac fermion F with B − L charge +1 for the left-handed component and −Q for the right-handed component.In the scalar sector, this model introduces one scalar doublet η, two scalar singlets χ and σ with B − L charge Q − 1, Q − 1 and Q + 1 respectively.Owing to newly introduced chiral fermions being singlets under the SM gauge group, one only need to check the [U (1) B−L ] × [Gravity] 2and [U (1) B−L ] 3 anomaly free conditions[43] free parameters: the new gauge coupling g , the new gauge boson mass M Z , the DM mass M DM and ν R 's B − L charge Q.Before investigate the DM phenomenon, we consider possible constraints from collider signature in Sec.III and dark radiation in Sec.IV.III.COLLIDER SIGNATURE Spontaneous breaking of the U (1) B−L gauge symmetry by the VEV of σ induces the mass term of the U (1) B−L gauge boson as [59] parton distribution function.As suggested by Ref.[53], a factor of k = 1.20 is also multiplied in order to include the QCD corrections.The results are shown in Fig.2.Clearly, for a fixed Q-value, a larger g leads to a larger cross section (shown in Fig.2 (a)).Meanwhile, for a fixed g -value, a larger |Q| leads to a smaller cross section (shown in Fig.2 (b)), mainly due to the suppression of BR(Z → + − ).
for fermion DM, actually leads to a smaller spin-independent

Fig. 10 . 4 FIG. 11 .
Fig. 10.It is obvious that even with Breit-Wigner enhancement, the annihilation cross sections are still several orders of magnitudes lower than future CTA limits.In left panel of Fig. 10, we show the impact of B −L charge Q. Off the resonance region, one has a similar cross section for Q = − 1 4 and Q = 3 2 , while the cross section is much smaller for Q = −4 due to sizable suppression of τ + τ − branching ratio.When the gauge boson Z is almost on-shell, the cross section could reach about 2 × 10 −30 cm 3 s −1 , 1 × 10 −30 cm 3 s −1 and 1 × 10 −31 cm 3 s −1 for Q = − 1 4 , Q = −4 and Q = 3 2 respectively.In right panel of Fig. 10, we

FIG. 14 . 1 1. 2 FIG. 15 .
FIG. 14. Scanning results for scalar DM H 0 1 .(a) Viable parameter space for relic density in the g -M Z plane.The solid, dotted and dashed lines (from the bottom up) correspond to LHC@13TeV exclusion limits for = − 1 4 , 3 2 and −4 respectively.(b) Spin-independent cross section as a function of M H 0 1 .(c) Annihilation cross section into τ + τ − final states as a function of M H 0 1 .(d) Allowed parameter space after applying all constraints.The gray points are excluded.

TABLE I .
Particle contents and corresponding charge assignment.

TABLE II .
Branching ratio of Z for different value of Q.