Z ′ -portal right-handed neutrino dark matter in the minimal U(1) X extended Standard Model
aa r X i v : . [ h e p - ph ] F e b YGHP16-06 Z ′ -portal right-handed neutrino dark matterin the minimal U(1) X extended Standard Model Nobuchika Okada a and Satomi Okada b a Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL35487, USA b Graduate School of Science and Engineering, Yamagata University,Yamagata 990-8560, Japan
Abstract
We consider a concise dark matter (DM) scenario in the context of a non-exotic U(1) ex-tension of the Standard Model (SM), where a new U(1) X gauge symmetry is introduced alongwith three generation of right-handed neutrinos (RHNs) and an SM gauge singlet Higgs field.The model is a generalization of the minimal gauged U(1) B − L (baryon number minus leptonnumber) extension of the SM, in which the extra U(1) X gauge symmetry is expressed as a linearcombination of the SM U(1) Y and U(1) B − L gauge symmetries. We introduce a Z -parity andassign an odd-parity only for one RHN among all particles, so that this Z -odd RHN playsa role of DM. The so-called minimal seesaw mechanism is implemented in this model withonly two Z -even RHNs. In this context, we investigate physics of the RHN DM, focusing onthe case that this DM particle communicates with the SM particles through the U(1) X gaugeboson ( Z ′ boson). This “ Z ′ -portal RHN DM” scenario is controlled by only three free param-eters: the U(1) X gauge coupling ( α X ), the Z ′ boson mass ( m Z ′ ), and the U(1) X charge of theSM Higgs doublet ( x H ). We consider various phenomenological constraints to identify a phe-nomenologically viable parameter space. The most important constraints are the observed DMrelic abundance and the latest LHC Run-2 results on the search for a narrow resonance withthe di-lepton final state. We find that these are complementary with each other and narrowthe allowed parameter region, leading to the lower mass bound of m Z ′ & . . INTRODUCTION Neutrino masses and a suitable candidate for the dark matter are the major missing pieces inthe Standard Model (SM), which require us to extend the SM. The minimal B − L model [1–6]is a simple, well-motivated extension of the SM to incorporate the neutrino masses, where theglobal U(1) B − L (baryon number minus lepton number) symmetry in the SM is gauged. In thepresence of the three right-handed neutrinos (RHNs) the model is free from all the gauge andgravitational anomalies. Associated with the spontaneous B − L gauge symmetry breaking bya vacuum expectation value (VEV) of the B − L Higgs field, the RHNs and the B − L gaugeboson ( Z ′ boson) acquire their masses. With the generated Majorana masses for the RHNs, theseesaw mechanism [7–11] is implemented, and the light SM neutrino mass is generated afterthe electroweak symmetry breaking. The mass spectrum of the new particles introduced inthe minimal B − L model (the Z ′ boson, the Majorana RHNs and the B − L Higgs boson) iscontrolled by the B − L gauge symmetry breaking scale. If the breaking scale lies around theTeV scale, the B − L model can be tested at the Large Hadron Collider (LHC).Among various possibilities, a concise way of introducing a dark matter (DM) candidate inthe minimal B − L model has been proposed in Ref. [12]. Instead of extending the minimalparticle content, a Z -parity is introduced and an odd-parity is assigned to only one RHN whileeven-parities are assigned to all the other fields. Hence, the parity-odd RHN serves as theDM. On the other hand, two parity-even RHNs account for the neutrino mass generation viathe seesaw mechanism. This system is nothing but the so-called minimal seesaw [14, 15], whichis the minimal setup to reproduce the observed neutrino oscillation data with a predictionof one massless neutrino as well as the observed baryon asymmetry in the universe throughleptogenesis [16].There are two ways for the RHN DM to communicate with the SM particles. One is throughtwo Higgs bosons, which are expressed as linear combinations of the SM Higgs and the B − L Higgs bosons after the breaking of the U(1) B − L and the electroweak gauge symmetries. TheDM phenomenology for this case has been analyzed in [12, 17, 18]. The other way is that theinteractions between the DM and the SM particles are mediated by the Z ′ boson. This classof DM scenario is called “ Z ′ -portal DM” and has been attracting a lot of attention recently.In the scenario, a DM particle is introduced along with an electric-charge neutral vector field( Z ′ boson) in an extension of the SM with the so-called Dark Sector [19–22] or new gaugeinteractions [23–49]. The mediator Z ′ boson allows us to investigate a variety of DM physics,such as the DM relic abundance and the direct/indirect DM search. A remarkable feature of the We can consider the Z -parity as an emergent global symmetry in the limit of vanishing Dirac Yukawacouplings [13]. Z ′ boson resonance search at the LHC is complementary to the cosmologicalobservations of the Z ′ -portal DM in identifying a phenomenologically viable parameter region.Recently, the minimal B − L model with the RHN DM has been investigated in the light ofthe LHC Run-2 results [40]. Here, the RHN DM communicates with the SM particles mainlythrough the Z ′ gauge boson, and hence it is the Z ′ -portal DM scenario. In the model, the DMphysics is controlled by only two free parameters, the B − L gauge coupling and the Z ′ bosonmass. It has been found that the constraint from the observed DM relic abundance leads toa lower bound on the gauge coupling as a function of the Z ′ boson mass. On the other hand,the cross section of Z ′ boson production at the LHC is also determined by the same two freeparameters. The LHC Run-2 results on search for a narrow resonance with the di-lepton finalstates have been interpreted to obtain the upper bound on the gauge coupling as a function ofthe Z ′ boson mass. Combining the two results, an allowed parameter region has been identifiedto obtain the lower bound of m Z ′ & . B − L model to the so-called non-exotic U(1) X extension of the SM [50]. The non-exotic U(1) X model is the most general extension of the SMwith an extra anomaly-free U(1) gauge symmetry, which is described as a linear combinationof the SM U(1) Y and the U(1) B − L gauge groups. The particle content of the model is thesame as the one in the minimal B − L model except for the generalization of the U(1) X charge assignment for particles. Hence we can easily extend the minimal B − L model withthe RHN DM to the non-exotic U(1) X case. In this context, we perform detailed analysis toidentify a phenomenologically viable parameter region through the complementarity betweenthe DM physics and the LHC Run-2 results. Because of the U(1) X generalization, the Z ′ bosoncouplings with the SM particles are modified and the resultant parameter region is found to bequite different from the one obtained in Ref. [40]. For the LHC Run-2 results, we employ themost recent results reported by the ATLAS and the CMS collaborations in 2016 [51, 52].This paper is organized as follows. In the next section, we define the minimal non-exoticU(1) X extension of the SM with the Z ′ -portal RHN DM. In Sec. III, we analyze the DM relicabundance and identify a model parameter region to satisfy the observed DM relic abundance.In Sec. IV, we consider the results by the ATLAS and the CMS collaborations at the LHCRun-2 on the search for a narrow resonance with the di-lepton final states. We interpret theresults into the constraints on the Z ′ boson production in the minimal non-exotic U(1) X model.Combining all the constraints, we identify the allowed parameter regions in Sec. V. The lastsection is devoted to conclusions. 2 U(3) c SU(2) L U(1) Y U(1) X Z q iL / / x H + (1 / x Φ + u iR / / x H + (1 / x Φ + d iR − / − (1 / x H + (1 / x Φ + ℓ iL − / − / x H − x Φ + e iR − − x H − x Φ + H − / − / x H + N jR − x Φ + N R − x Φ − Φ x Φ +TABLE I. The particle content of the minimal U(1) X extended SM with Z parity. In addition to theSM particle content ( i = 1 , , N jR ( j = 1 ,
2) and N R ) and the U(1) X Higgs field(Φ) are introduced. Because of the Z parity assignment shown here, the N R is a unique (cold) DMcandidate. The extra U(1) X gauge group is defined with a linear combination of the SM U(1) Y andthe U(1) B − L gauge groups, and the U(1) X charges of fields are determined by two real parameters, x H and x Φ . Without loss of generality, we fix x Φ = 1 throughout this paper. II. THE MINIMAL NON-EXOTIC U(1) X MODEL WITH RHN DM
We first define our model by the particle content listed on Table I. The U(1) X gauge groupis identified with a linear combination of the SM U(1) Y and the U(1) B − L gauge groups, andhence the U(1) X charges of fields are determined by two real parameters, x H and x Φ . Notethat in the model the charge x Φ always appears as a product with the U(1) X gauge couplingand it is not an independent free parameter. Hence, we fix x Φ = 1 throughout this paper. Inthis way, we reproduce the minimal B − L model with the conventional charge assignment asthe limit of x H →
0. The limit of x H → + ∞ ( −∞ ) indicates that the U(1) X is (anti-)alignedto the U(1) Y direction. The anomaly structure of the model is the same as the minimal B − L model and the model is free from all the gauge and the gravitational anomalies in the presenceof the three RHNs. The introduction of the Z -parity is crucial to incorporate a DM candidatein the model while keeping the minimality of the particle content. The conservation of the Z -parity ensures the stability of the Z -odd RHN, and therefore it is a unique DM candidatein the model.The Yukawa sector of the SM is extended to have L Y ukawa ⊃ − X i =1 2 X j =1 Y ijD ℓ iL HN jR − X k =1 Y kN Φ N k CR N kR − Y N Φ N CR N R + h . c ., (1)where the first term is the neutrino Dirac Yukawa coupling, and the second and third terms arethe Majorana Yukawa couplings. Without loss of generality, the Majorana Yukawa couplings are3lready diagonalized in our basis. Note that because of the Z -parity, only the two generationRHNs are involved in the neutrino Dirac Yukawa coupling. Once the U(1) X Higgs field Φdevelops a nonzero VEV, the U(1) X gauge symmetry is broken and the Majorana mass termsfor the RHNs are generated. Then, the seesaw mechanism is automatically implemented in themodel after the electroweak symmetry breaking. Because of the Z -parity, only two generationRHNs are relevant to the seesaw mechanism. Even with two RHNs, the Yukawa couplingconstants Y ijD and Y kN posses the degrees of freedom large enough to reproduce the neutrinooscillation data with a prediction of one massless eigenstate. The baryon asymmetry in theuniverse can also be reproduced with the two RHNs [15] (see, for example, Ref. [53] for detailedanalysis of leptogenesis at the TeV scale with two RHNs).The renormalizable scalar potential for the SM Higgs doublet ( H ) and the U(1) X Higgsfields is given by V = λ H (cid:18) H † H − v (cid:19) + λ Φ (cid:18) Φ † Φ − v X (cid:19) + λ mix (cid:18) H † H − v (cid:19) (cid:18) Φ † Φ − v X (cid:19) , (2)where all quartic couplings are chosen to be positive. At the potential minimum, the Higgsfields develop their VEVs as h H i = v √ ! , h Φ i = v X √ . (3)In this paper, we assume λ mix ≪
1, so that the mixing between the SM Higgs boson andthe U(1) X Higgs boson are negligibly small. Hence, the RHN DM communicates with theSM particles only through the Z ′ boson. Associated with the U(1) X symmetry breaking, theMajorana neutrinos N jR ( j = 1 , N R and the Z ′ gauge boson acquire theirmasses as m jN = Y jN √ v X , m DM = Y N √ v X , m Z ′ = g X r v X + v ≃ g X v X , (4)where g X is the U(1) X gauge coupling, and we have used the LEP constraint [54, 55] v X ≫ v .Because of the LEP constraint, the mass mixing of the Z ′ boson with the SM Z boson is verysmall, and we neglect it in our analysis in this paper.Assuming λ mix ≪
1, we focus on the Z ′ -portal nature of the RHN DM. In this case, only fourfree parameters ( g X , m Z ′ , m DM , and x H ) are involved in our analysis. As we will discuss in thenext section, it turns out that the condition of m DM ≃ m Z ′ / m DM does not work as an independent parameter, sothat our results are described by only three free parameters. This assumption is, in fact, not essential. When λ mix is sizable, the RHN DM can communicate with theSM particles also through the Higgs bosons. This so-called Higgs portal RHN DM case has been analyzedin [12, 17, 18] and it has been shown that the RHN DM mass is required to be close to a half of either oneof the Higgs boson masses in order to reproduce the observed relic abundance. Such a parameter region isdistinguishable from that in our Z ′ -portal RHN DM case, and we can investigate the two cases separately. II. COSMOLOGICAL CONSTRAINTS ON Z ′ -PORTAL RHN DM. In the Planck satellite experiments, the DM relic abundance is measured at the 68% limitas [56] Ω DM h = 0 . ± . . (5)In this section, we evaluate the DM relic abundance and identify an allowed parameter regionto satisfy the upper bound of Ω DM h ≤ . dYdx = − xs h σv i H ( m DM ) (cid:0) Y − Y EQ (cid:1) , (6)where the temperature of the universe is normalized by the mass of the RHN DM as x = m DM /T , H ( m DM ) is the Hubble parameter at T = m DM , Y is the yield (the ratio of the DMnumber density to the entropy density s ) of the RHN DM, Y EQ is the yield of the DM particle inthermal equilibrium, and h σv i is the thermal average of the DM annihilation cross section timesrelative velocity ( v ). Explicit formulas of the quantities involved in the Boltzmann equationare as follows: s = 2 π g ⋆ m DM x , H ( m DM ) = r π g ⋆ m DM M P , sY EQ = g DM π m DM x K ( x ) , (7)where M P = 2 . × GeV is the reduced Planck mass, g DM = 2 is the number of degreesof freedom for the DM particle, g ⋆ is the effective total number of degrees of freedom for theparticles in thermal equilibrium (in the following analysis, we use g ⋆ = 106 .
75 for the SMparticles), and K is the modified Bessel function of the second kind. In our Z ′ -portal DMscenario, a DM pair annihilates into the SM particles through the Z ′ boson exchange in the s -channel. The thermal average of the annihilation cross section is given by h σv i = ( sY EQ ) − g DM m DM π x Z ∞ m DM ds ˆ σ ( s ) √ sK (cid:18) x √ sm DM (cid:19) , (8)where ˆ σ ( s ) = 2( s − m DM ) σ ( s ) is the reduced cross section with the total annihilation crosssection σ ( s ), and K is the modified Bessel function of the first kind. The total cross section ofthe DM pair annihilation process N N → Z ′ → f ¯ f ( f denotes the SM fermions) is calculatedas σ ( s ) = π α X p s ( s − m DM )( s − m Z ′ ) + m Z ′ Γ Z ′ F ( x H ) , (9)where F ( x H ) = 13 + 16 x H + 10 x H = 10 ( x H + 0 . + 6 . , (10)5
940 1950 1960 1970 19800.110.120.130.140.150.16 m DM @ GeV D W h m DM @ GeV D W h FIG. 1. The relic abundance of the Z ′ -portal RHN DM as a function of its mass ( m DM ) for m Z ′ = 4TeV. In the left panel, we have fixed x H = 0 (the minimal B − L model limit) and shown the relicabundance for various values of the gauge coupling, α X = 0 . . .
028 and 0 .
030 (solid linesfrom top to bottom). In the right panel, we have fixed α X = 0 .
027 and shown the relic abundance forvarious values of x H = − .
8, 0, 0 . . . ≤ Ω DM h ≤ . and the total decay width of Z ′ boson is given byΓ Z ′ = α X m Z ′ " F ( x H ) + (cid:18) − m DM m Z ′ (cid:19) θ (cid:18) m Z ′ m DM − (cid:19) . (11)Here, we have neglected all SM fermion masses and assumed m jN > m Z ′ /
2, for simplicity.Now we solve the Boltzmann equation numerically, and find the asymptotic value of theyield Y ( ∞ ) to evaluate the present DM relic density asΩ DM h = m DM s Y ( ∞ ) ρ c /h , (12)where s = 2890 cm − is the entropy density of the present universe, and ρ c /h = 1 . × − GeV/cm is the critical density. Our analysis involves four parameters, namely α X = g X / (4 π ), m Z ′ , m DM and x H . For m Z ′ = 4 TeV, we show in Fig. 1 the resultant DM relic abundanceas a function of the DM mass, along with the range of the observed DM relic abundance,0 . ≤ Ω DM h ≤ . x H = 0, which is the minimal B − L model limit. The solid lines from top to bottom showthe resultant DM relic abundances for various values of the gauge coupling, α X = 0 . . .
028 and 0 . α X ≥ .
027 for m Z ′ = 4 TeV and x H = 0in order to be able to reproduce the observed relic abundance. In addition, we can see thatthe enhancement of the DM annihilation cross section via the Z ′ boson resonance is necessaryto satisfy the cosmological constraint and hence, m DM ≃ m Z ′ /
2. The right panel shows our6 .0 2.5 3.0 3.5 4.0 4.5 5.00.00100.01000.00500.00200.00300.00150.01500.0070 m Z ' @ TeV D Α X FIG. 2. The lower bounds on α X as a function of m Z ′ for various values of x H , to satisfy thecosmological constraint of 0 . ≤ Ω DM h ≤ . x H = −
3, +1, −
2, 0 and −
1, respectively. As the input x H value is going away from the point of x H = − .
8, the lower bound on α X is increasing. results for various values of x H with the fixed α X = 0 . x H = − .
8, 0, 0 . .
0, respectively. From Eqs. (8)-(11), we can see that the DM annihilation cross section for m DM ≃ m Z ′ / /F ( x H ). Therefore, the maximum annihilation cross section for the fixed values of α X , m Z ′ and m DM ≃ m Z ′ / x H = − .
8. Since the function F ( x H ) is symmetric about thepoint of x H = − .
8, the results shown in the left panel indicate the constraint − . ≤ x H ≤ m Z ′ = 4 TeV and α X = 0 . α X as a function of m Z ′ for various values of x H , toreproduce the observed DM relic abundance in the range of 0 . ≤ Ω DM h ≤ . x H = −
3, +1, −
2, 0 and −
1, respectively. Forfixed α X and m Z ′ , the DM annihilation cross section becomes maximum for x H ≃ − . Z ′ boson decay width. As an input x H value is going away from the point of x H = − .
8, the decay width becomes larger and the DM annihilation cross section is reducing.As a result, the lower bound on the gauge coupling is increasing.
IV. LHC RUN-2 CONSTRAINTS
In 2015, the LHC Run-2 started its operation with a 13 TeV collider energy. The most recentresults by the ATLAS and the CMS collaborations with the combined 2015 and 2016 data were7eported at the ICHEP 2016 conference. The ATLAS and the CMS collaborations continuetheir search for Z ′ boson resonance with di-lepton final states at the LHC Run-2. Their resultshave shown significant improvements for the upper limits of the Z ′ boson production crosssection [51, 52] from those obtained by the LHC Run-1 [57, 58]. In this section, we will employthe most recent LHC Run-2 results to derive LHC constraints on the model parameters, α X , m Z ′ and x H .Let us calculate the cross section for the process pp → Z ′ + X → ℓ + ℓ − + X . The differentialcross section with respect to the invariant mass M ℓℓ of the final state di-lepton is given by dσdM ℓℓ = X q, ¯ q Z M ℓℓE dx M ℓℓ xE f q ( x, Q ) f ¯ q (cid:18) M ℓℓ xE , Q (cid:19) ˆ σ ( q ¯ q → Z ′ → ℓ + ℓ − ) , (13)where f q is the parton distribution function for a parton (quark) “ q ”, and E CM = 13 TeV is thecenter-of-mass energy of the LHC Run-2. In our numerical analysis, we employ CTEQ6L [59]for the parton distribution functions with the factorization scale Q = m Z ′ . Here, the crosssection for the colliding partons is given byˆ σ ( q ¯ q → Z ′ → ℓ + ℓ − ) = π α X M ℓℓ ( M ℓℓ − m Z ′ ) + m Z ′ Γ Z ′ F qℓ ( x H ) , (14)where the function F qℓ ( x H ) is given by F uℓ ( x H ) = (8 + 20 x H + 17 x H )(8 + 12 x H + 5 x H ) ,F dℓ ( x H ) = (8 − x H + 5 x H )(8 + 12 x H + 5 x H ) (15)for q being the up-type ( u ) and down-type ( d ) quarks, respectively. By integrating the differ-ential cross section over a range of M ℓℓ set by the ATLAS and the CMS analysis, respectively,we obtain the cross section to be compared with the upper bounds obtained by the ATLASand the CMS collaborations.In the analysis by the ATLAS and the CMS collaborations, the so-called sequential SM Z ′ ( Z ′ SSM ) model [61] has been considered as a reference model. We first analyze the sequential Z ′ model to check a consistency of our analysis with the one by the ATLAS collaboration.In the sequential Z ′ model, the Z ′ SSM boson has exactly the same couplings with quarks andleptons as the SM Z boson. With the couplings, we calculate the cross section of the process pp → Z ′ SSM + X → ℓ + ℓ − + X like Eq. (13). By integrating the differential cross section inthe region of 128 GeV ≤ M ℓℓ ≤ Z ′ SSM boson mass. Our result is shown as a solid line inthe left panel on Fig. 3, along with the plot presented by the ATLAS collaboration [51, 60]. Inthe ATLAS paper [51], the lower limit of the Z ′ SSM boson mass is found to be 4 .
05 TeV, which8 - m Z ' SSM @ TeV D Σ B @ pb D - m Z ' @ TeV D Σ B @ pb D FIG. 3. Left panel: the cross section as a function of the Z ′ SSM mass (solid line) with k = 1 . α X with k = 1 .
28, for theminimal B − L model limit ( x H = 0). The solid lines from left to right correspond to α X = 10 − ,10 − . , 10 − , 10 − . , 10 − , 10 − . , 10 − , and 10 − . , respectively. is read off from the intersection point of the theoretical prediction (diagonal dashed line) andthe experimental cross section bound (lower horizontal solid curve (in red)). Here, we havealso shown the plot presented in Ref. [60] (upper horizontal solid curve (in red)). We can seethe dramatic improvement from the 2015 results [60] to the 2016 results [51]. In order to takeinto account the difference of the parton distribution functions used in the ATLAS and ouranalysis and QCD corrections of the process, we have scaled our resultant cross section by afactor k = 1 .
28, with which we can obtain the same lower limit of the Z ′ SSM boson mass as 4 . k = 1 .
28 (solid line) is very consistent withthe theoretical prediction (diagonal dashed line) presented in Ref. [51]. This factor is used inour analysis of the Z ′ boson production process in the following.Now we calculate the cross section of the process pp → Z ′ + X → ℓ + ℓ − + X for variousvalues of α X , m Z ′ and x H . For x H = 0 (the minimal B − L model limit), we show our results inthe right panel of Fig. 3, along with the plots in the ATLAS papers [51, 60]. The diagonal solidlines from left to right correspond to α X = 10 − , 10 − . , 10 − , 10 − . , 10 − , 10 − . , 10 − , and10 − . . From the intersections of the lower horizontal curve (in red) and diagonal solid lines,we can read off the lower bounds on the Z ′ boson mass for the corresponding α X values. Forexample, m Z ′ > . α X = 0 . α X as a function of the Z ′ boson mass. For various values of x H we do the same analysis andfind the upper bound.We apply the same strategy and compare our result for the Z ′ SSM model with the one bythe CMS collaboration [52, 62]. According to the CMS analysis, we integrate the differential9 .5 2.0 2.5 3.0 3.5 4.0 4.51 ´ - ´ - ´ - ´ - ´ - m Z ' SSM @ TeV D Σ H pp ® Z ' + X ® ll + X L (cid:144) Σ H pp ® Z + X ® ll + X L ´ - ´ - ´ - ´ - ´ - ´ - ´ - m Z ' @ TeV D Σ H pp ® Z ' + X ® ll + X L (cid:144) Σ H pp ® Z + X ® ll + X L FIG. 4. Left panel: the cross section ratio as a function of the Z ′ SSM mass (solid line) with k = 1 . α X with k = 1 .
61 for x H = 0. The solid lines from left to right correspond to α X = 10 − . , 10 − , 10 − . , 10 − , 10 − . , 10 − ,and 10 − . , respectively. - - x H m Z ' @ T e V D (cid:144) g X FIG. 5. The lower bound on m Z ′ /g X as a function of x H . We have employed the final LEP 2 data [55]at 95% confidence level. cross section in the range of 0 . ≤ M ℓℓ /m Z ′ SSM ≤ .
05. In the CMS analysis, a limit has beenset on the ratio of the Z ′ SSM boson cross section to the
Z/γ ∗ cross section in a mass windowof 60 to 120 GeV, which is predicted to be 1928 pb. Our result is shown as a diagonal solidline in the left panel of Fig. 4, along with the plot presented in Ref. [52]. The analysis in thisCMS paper leads to the lower limit of the Z ′ SSM boson mass as 4 . .0 2.5 3.0 3.5 4.0 4.5 5.010 - - m Z ' @ TeV D Α X - - m Z ' @ TeV D Α X FIG. 6. Left panel: the upper bounds on α X as a function of m Z ′ for x H = −
1, 0 and +1 from topto bottom for both of the solid and dashed lines, respectively. The solid lines denote the bounds fromthe ATLAS results [51] while the dashed lines denote the bounds from the CMS results [52]. Rightpanel: the upper bounds on α X after combining the ATLAS and the CMS results shown in the leftpanel. The solid lines correspond to the combined upper bounds for x H = −
1, 0 and +1 from top tobottom, respectively. The perturbativity bounds of Eq. (17) for x H = −
1, 0 and +1 are shown as thehorizontal dashed-dotted lines from top to bottom, respectively. cross section bound (lower horizontal solid curve (in red)). Here, we have also shown the plotpresented in Ref. [62] (upper horizontal solid curve (in red)). As in the left panel of Fig. 3,we can see the dramatic improvement from the 2015 results [62] to the 2016 results [52]. Inorder to obtain the same lower mass limit of m Z ′ SSM ≥ . k = 1 .
61. We can see that our result (solid line) are very consistent with the theoretical crosssection (dashed line) presented in Ref. [52].With the factor of k = 1 .
61, we have calculated the cross section of the process pp → Z ′ + X → ℓ + ℓ − + X for various values of α X , m Z ′ and x H . For the minimal B − L model limit,we show our results in the right panel of Fig. 4, along with the plots in the CMS papers [52, 62].The diagonal solid lines from left to right correspond to α X = 10 − . , 10 − , 10 − . , 10 − , 10 − . ,10 − , and 10 − . . From the intersections of the lower horizontal curve and the diagonal solidlines, we can read off the lower bounds on the Z ′ boson mass for the corresponding α X values.For example, m Z ′ > . α X = 10 − . . In this way, we have obtained the upper boundon α X as a function of the Z ′ boson mass. For various values of x H we do the same analysisand find the upper bound.The search for effective 4-Fermi interactions mediated by a Z ′ boson at the LEP leads to alower bound on m Z ′ /g X [54, 55]. Employing the limits from the final LEP 2 data [55] at 95%confidence level, we follow Ref. [63] and derive a lower bound on m Z ′ /g X as a function of x H .11ur result is shown in Fig. 5. For example, we find m Z ′ g X ≥ .
94 TeV . (16)for the minimal B − L model limit, which is consistent with the result found in Ref. [64]. Wefind that for any values of x H , the LEP constraints are always weaker than the LHC Run-2constraints for m Z ′ ≤ X gauge couplingto avoid the Landau pole in its renormalization group evolution α X ( µ ) up to the Plank mass,1 /α X ( M P l ) >
0, where M P l = 1 . × GeV. Let us define the gauge coupling α X usedin our analysis for the dark matter physics and LHC physics as the running gauge coupling α X ( µ ) at µ = m Z ′ . Employing the renormalization group equation at the one-loop level with m N = m N = m Φ = m Z ′ , for simplicity, we find α X < πb X ln h M Pl m Z ′ i , (17)where b X = (72 + 64 x H + 41 x H ) / α X as a function of m Z ′ for x H = −
1, 0 and+1. In the left panel, the solid (dashed) lines from top to bottom denote the upper boundson α X for x H = −
1, 0 and +1, respectively, obtained from the ATLAS results [51] (the CMSresults [52]). For m Z ′ . − . x H = −
1, 0 and +1 from top to bottom, respectively. The perturbativity bounds of Eq. (17)for x H = −
1, 0 and +1 are shown as the horizontal dashed-dotted lines from top to bottom,respectively.
V. COMPLEMENTARITY BETWEEN THE COSMOLOGICAL AND THE LHCCONSTRAINTS
Now we combine the constraints that we have obtained in the previous two sections. TheRHN DM abundance has led to the lower bound on the U(1) X gauge coupling for fixed m Z ′ and x H , while the upper limit on the production cross section of the Z ′ boson at the LHC has derivedthe upper bound on the gauge coupling. Therefore, the two constraints are complementary toeach other and, once combined, the model parameter space is more severely constrained.We show the results for various x H values in Fig. 7. The top-left panel shows the results forthe minimal B − L model limit ( x H = 0). The (black) solid line shows the lower bound on α X .0 2.5 3.0 3.5 4.0 4.5 5.01 ´ - ´ - m Z ' @ TeV D Α X ´ - ´ - m Z ' @ TeV D Α X ´ - ´ - m Z ' @ TeV D Α X ´ - ´ - ´ - m Z ' @ TeV D Α X FIG. 7. Allowed parameter region for the Z ′ -portal RHN DM scenario. The top-left panel shows theresults for the minimal B − L model limit ( x H = 0). The (black) solid line denotes the lower boundon α X as a function of m Z ′ to reproduce the observed DM relic abundance. The lower dashed line(in red) shows the upper bound on α X obtained from the search results for Z ′ boson resonance at theLHC. The shaded region is the final result after combining the cosmological and the LHC constraints,leading to the lower mass bound of m Z ′ & . α X asthe dashed-dotted line. The top-right, the bottom-left and the bottom-right panels are same as thetop-left panel, but x H = − − as a function of m Z ′ to reproduce the observed DM relic abundance. The lower dashed line (inred) shows the upper bound on α X obtained from the search results for Z ′ boson resonance bythe ATLAS [51] and the CMS [52] collaborations. Here, the ATLAS and the CMS bounds arecombined as in the right panel on Fig. 6. The shaded region is the final result after combiningthe cosmological and the LHC constraints, leading to the lower mass bound of m Z ′ & . m Z ′ & . α X from the LEP constraint in Eq. (16) is depicted as the dotted line, which turns out to be13 - - - - x H Α X FIG. 8. Allowed parameter region for the Z ′ -portal RHN DM scenario for m Z ′ = 4 TeV. The (black)solid line shows the cosmological lower bound on α X as a function of x H . The dashed line (in red)shows the upper bound on α X obtained from the combined ATLAS and CMS bounds. The shadedregion is the final result for the allowed parameter space after combining the cosmological and theLHC constraints, leading to the allowed range of − . ≤ x H ≤ .
3. The LEP bound appears abovethe plot range. The dashed-dotted line denotes the theoretical upper bound on α X in Eq. (17). weaker than the LHC bound. We also show the theoretical upper bound on α X in Eq. (17)as the dashed-dotted line. If we impose this bound, it provides the most severe upper boundfor the range of 4 . . m Z ′ . . x H = − − x H ≃ −
1, while no allowed region has beenfound for a x H value outside the range of − . ≤ x H ≤ m Z ′ = 4 TeV, we show the allowed parameter region in Fig. 8. The(black) solid line shows the lower bound on α X as a function of x H to reproduce the observedDM relic abundance. As discussed in Sec. III, the minimum α X appears at x H ≃ − . α X obtained from the combined ATLASand CMS constraints. The shaded region is the final result for the allowed parameter spaceafter combining the cosmological and the LHC constraints, leading to the allowed range of − . ≤ x H ≤ .
3. The LEP upper bound appears above the plot range. The dashed-dottedline denotes the theoretical upper bound from the perturbativity of the running α X ( µ ) up tothe Planck scale.The maximum value of α X to satisfy the LHC bound appears at x H ≃ −
1. This means thatthe cross section of the Z ′ boson production at the LHC exhibits its minimum at x H ≃ − Z ′ boson is very narrow, we approximate Eq. (14) asˆ σ ( q ¯ q → Z ′ → ℓ + ℓ − ) ≃ π α X M ℓℓ (cid:20) πm Z ′ Γ Z ′ δ ( M ℓℓ − m Z ′ ) (cid:21) F qℓ ( x H ) ∝ F qℓ ( x H ) F ( x H ) . (18)Using the explicit formulas for F ( x H ) and F qℓ ( x H ) given in Eqs. (10) and (15), we can verifythat the function F qℓ ( x H ) /F ( x H ) exhibits a minimum at x H ≃ − VI. CONCLUSIONS
We have considered the minimal non-exotic U(1) X extension of the SM, which is free fromall the gauge and the gravitational anomalies in the presence of three right-handed neutrinos.After the breaking of the U(1) X and the electroweak gauge symmetries, the SM neutrino massesand flavor mixings are generated through the seesaw mechanism. We have extended this modelby introducing a Z -parity and assigned an odd-parity to one RHN while even-parities to all theother particles. Thanks to the parity, the Z -odd RHN is stable and hence the DM candidate.No extension of the minimal particle content is necessary to incorporate a DM candidate intothe model. With the other two RHNs, the seesaw mechanism works to account for the neutrinooscillation data with one massless neutrino. In this model, the RHN DM communicates withthe SM particles through the Z ′ boson exchange. We have investigated this Z ′ -portal RHNDM scenario in this paper.Phenomenology of the scenario is controlled by only four free parameters, namely, the U(1) X gauge coupling ( α X ), the RHN DM mass ( m DM ), the Z ′ boson mass ( m Z ′ ) and the U(1) X chargeof the SM Higgs doublet field ( x H ). We have first considered the cosmological constraint of thescenario. In order to reproduce the observed DM relic density, we have found it necessary toenhance the DM annihilation cross section via Z ′ boson resonance. Therefore, the RHN DMmass is always set to be m DM ≃ m Z ′ /
2, and the number of the free parameters is reducedto three. The three parameters are constrained by the DM relic abundance. For example,the lower bound on α X has been obtained as a function of m Z ′ for a fixed x H . We have nextconsidered the LHC constraints on the Z ′ boson production cross section by employing the mostrecent results by the ATLAS and the CMS collaborations on the search for a narrow resonancewith the di-lepton final state. We have derived the lower bound on α X as a function of m Z ′ for a fixed x H . In constraining the model parameter space, the cosmological and the LHCbounds are complementary with each other, and we have narrowed the phenomenologicallyviable parameter region by combining them. For example, we have found the lower limit of the Z ′ boson mass to be m Z ′ & . X symmetry breaking scaleand the perturbativity bound on the running U(1) X gauge coupling below the Planck scale. ACKNOWLEDGMENTS
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