Dark matter and muon (g−2) in local U(1 ) L μ − L τ -extended Ma Model
KKIAS-P15054
Dark matter and muon ( g − in local U (1) L µ − L τ -extended MaModel Seungwon Baek ∗ School of Physics, KIAS, 85 Hoegiro, Seoul 02455, Korea
We consider right-handed neutrino dark matter N in local U (1) L µ − L τ -extendedMa model. With the light U (1) µ − τ gauge boson ( m Z (cid:48) ∼ O (100) MeV) and small U (1) µ − τ gauge coupling ( g Z (cid:48) ∼ − − − ) which can accommodate the muon( g −
2) anomaly and is still allowed by other experimental constraints, we show thatwe can get correct relic density of dark matter for wide range of dark matter mass( M ∼ −
100 GeV), although the gauge coupling constant g Z (cid:48) is small. This isdue to the fact that the annihilation cross section of dark matter pair is enhancedby M /m Z (cid:48) in the processes N N → Z (cid:48) Z (cid:48) or N N → Z (cid:48) H . We also consider theconstraints from direct detection, collider searches. PACS numbers: ∗ Electronic address: [email protected] a r X i v : . [ h e p - ph ] O c t I. INTRODUCTION
About 27% of the universe is composed of dark matter, but we do not know its natureyet. We may, however, find a clue for the dark matter in other sector of the standard model(SM), such as neutrino sector. One example is the models where the neutrino masses aregenerated radiatively with dark matter as an essential component [1].In Ref. [2], we extended Ma’s scotogenic model [3] so that the model has gauged L µ − L τ symmetry. In fact, three symmetries L e − L µ , L e − L τ , and L µ − L τ , where L i is thelepton number associated with the flavor i , can be gauged without the extension of theSM particle content . The gauge anomaly cancels between different generations. In thatpaper we demonstrated that the neutrino mass matrix has two-zero texture due to the gaugesymmetry, making the theory very predictive. Especially we predicted the neutrino masseshave inverted hierarchy and the Dirac CP phase is close to maximal ( ∼ ◦ ).In this paper we consider the dark matter phenomenology of the model. Especially wewill show that we can get correct dark matter relic abundance and explain the muon ( g − g − µ ) anomaly at the same time. According to [4], almost all the region which can explain( g − µ is excluded by the neutrino trident production in U (1) µ − τ model. However, the regionfor Z (cid:48) mass, m Z (cid:48) (cid:46)
400 MeV, and for the extra U (1) gauge coupling, g Z (cid:48) ∼ × − − − ,is still allowed and can accommodate ( g − µ anomaly. In this paper we concentrate onthis region, since the current experimental results still show 3-4 σ deviation from the SMpredictions.The analysis in this paper is applicable to more general dark matter models with light Z (cid:48) gauge boson coupled to right-handed neutrinos where the lightest right-handed neutrinois the dark matter candidate. For example, the inert doublet scalar in the Ma model isirrelevant for our discussion on dark matter and we would get similar results with this paperif only the right-handed neutrinos have similar structure.This paper is organized as follows. In Section II, we briefly review our model in theprospect of dark matter phenomenology. In Section III, we show numerical results. InSection IV, we conclude. We will denote L µ − L τ as just µ − τ for notational simplicity. L e L µ L τ e cR µ cR τ cR N ce N cµ N cτ Φ η SSU (2) L U (1) Y − / / / U (1) L µ − L τ − − − Z + + − + − +TABLE I: The particle content and the charge assignment under SU (2) L × U (1) Y × U (1) L µ − L τ × Z . II. THE MODEL
The original Ma model [3] introduces right-handed neutrinos N ci ( i = e, µ, τ ), and SU (2) L -doublet scalar η , both of which are odd under discrete symmetry Z . As a consequence thelightest state of them do not decay into the standard model (SM) particles and can be adark matter candidate. The Yukawa interactions involving L, N c , η fields in the original Mamodels are given by L = − M ij N ci N cj − y ij Φ † L i e cj + f ij η · L i N cj , (1)where Φ is the SM Higgs doublet field and η · L i ≡ (cid:15) ab η a L ib in SU (2) L space. The neutrinomass terms come from one one-loop diagrams involving both N ci and η [3].To extend the Ma model to local U (1) µ − τ symmetry, we just need to introduce one addi-tional scalar particle S charged under U (1) µ − τ to break the abelian symmetry spontaneously.The particle content and the charge assignment under SU (2) L × U (1) Y × U (1) µ − τ × Z areshown in Table I.The new gauge interactions are dictated by the gauge covariant derivative to give∆ L = (cid:88) ψ = l fL ,e fR ,N fR g Z (cid:48) Q (cid:48) ψ ψγ µ Z (cid:48) µ ψ, (2)where f = µ, τ .Due to U (1) µ − τ symmetry all the terms in (1) are not allowed. And the Yukawa interac-tion and right-handed neutrino mass terms become more restricted to be L = − M ee N ce N ce − M µτ ( N cµ N cτ + N cτ N cµ ) − h eµ ( N ce N cµ + N cµ N ce ) S − h eτ ( N ce N cτ + N cτ N ce ) S ∗ + η · ( f e L e N ce + f µ L µ N cµ + f τ L τ N cτ ) − Φ † ( y e L e e cR + y µ L µ µ cR + y τ L τ τ cR )+ h.c, (3)where all the fermions are Weyl spinors. After S gets vev v S ( (cid:104) S (cid:105) = v S / √ M R = M ee h eµ v S h eτ v S h eµ v S M µτ e iθ R h eτ v S M µτ e iθ R . (4)By appropriate phase rotation, we can make all the parameters real except the one in (2 , θ R . The matrix M R is symmetric and canbe diagonalized by a unitary matrix V T M R V = diag( M , M , M ) . (5)The scalar potential of Φ, η , and S is given by V = µ | Φ | + µ η | η | + µ S | S | + 12 λ | Φ | + 12 λ | η | + λ | Φ | | η | + λ | Φ † η | + 12 λ (cid:104) (Φ † η ) + h.c. (cid:105) + 12 λ | S | + λ | Φ | | S | + λ | η | | S | . (6)After Φ and S get vev, v and v S , respectively, we can writeΦ = √ ( v + h ) , S = 1 √ v S + s ) , (7)in the unitary gauge. Then the two neutral states h and s can mix with each other withmixing angle α , whose mass eigenstates we denote as H and H with masses m H and m H ,respectively [5]. Here H is the SM-like Higgs boson with m H ≈
125 GeV. In this paperwe will assume this “Higgs portal” term, i.e. the λ , is small, because its mixing angle isstrongly suppressed by the study of Higgs signal strength [5]. III. MUON ( g − , RELIC DENSITY, DIRECT DETECTION OF DARKMATTER, AND OTHER TESTS OF THE MODEL In this section we concentrate on the dark matter phenomenology, especially the relicdensity and the direct detection, of the model in the region which can explain the muon( g −
2) anomaly. Let us first consider the muon ( g −
2) in our model. The discrepancybetween experimental measurement [6] and the SM prediction [7]∆ a µ ≡ a exp µ − a SM µ = (295 ± × − , (8)is about 3 . σ and can be explained by the U (1) µ − τ gauge boson contribution [8, 9]. Althoughthe neutrino trident production process disfavors the Z (cid:48) explanation of muon ( g −
2) for m Z (cid:48) (cid:38) . Z (cid:48) region is still consistent with ( g − µ .According to the Ref. [4], the allowed region for ( g − µ is characterized by light Z (cid:48) , m Z (cid:48) (cid:46) . Z (cid:48) gauge coupling constant, 10 − (cid:46) g Z (cid:48) (cid:46) − . For this smallgauge coupling constant, it is naively expected the annihilation processes of the dark matterpair at the electroweak scale dominated by [10] N N → Z (cid:48)∗ → l + l − , ν l ν l ( l = µ, τ ) ,N N → Z (cid:48) Z (cid:48) , (9)would have very small cross sections. As a consequence, the dark matter relic density wouldoverclose the universe. It turns out that this is not the case.The dominant dark matter annihilation processes in our region of interest ( i.e. light Z (cid:48) and small g Z (cid:48) ) are N N → Z (cid:48) Z (cid:48) , and N N → Z (cid:48) H , (10)where H is the lighter mass eigenstate between the SM Higgs and the U (1) µ − τ breakingscalar. For the second process to occur, H should also be light enough to be kinematicallyallowed. The relevant diagrams are shown in Fig. 1.We notice that the longitudinal Z (cid:48) polarization has enhancement factor, (cid:15) ∗ µ ( p ) ∼ p µ /m Z (cid:48) ,when its energy is much larger than its mass. Since the total energy scale is almost fixed bythe dark matter mass in dark matter annihilation, there is an enhancement factor M /m Z (cid:48) for each Z (cid:48) in the external or internal line in the annihilation diagram. Consequently the N N Z ′ Z ′ ( d ) N i ( e ) ( f )( c ) H N i ( b ) N N H i Z ′ Z ′ N N H i Z ′ Z ′ ( a ) FIG. 1: Feynman diagrams for the processes, N N → Z (cid:48) Z (cid:48) and N N → Z (cid:48) H . Here H i ( i = 1 , N i ( i = 1 , ,
3) are three right-handed neutrino mass eigenstates. diagrams with two Z (cid:48) gauge boson lines are most enhanced. And the enhancement factorin the annihilation cross section is M /m Z (cid:48) . This large enhancement can compensate thesuppression due to small gauge coupling constant g Z (cid:48) allowed by the ( g − µ . For example,explicit calculation shows the annihilation cross section times relative velocity of the process, N N → Z (cid:48) Z (cid:48) , in Fig. 1 (a)-(c), is given by σv rel (cid:39) g Z (cid:48) v S M s πm Z (cid:48) (cid:104) h eµ (cid:61) ( V V ) + h eτ (cid:61) ( V V ) (cid:105) (cid:18) s α s − m H + c α s − m H (cid:19) + g Z (cid:48) M v πm Z (cid:48) (cid:0) | V | − | V | (cid:1) − √ g Z (cid:48) v S c α M sv πm Z (cid:48) ( s − m H ) (cid:0) | V | − | V | (cid:1) (cid:104) h eµ (cid:61) ( V V ) + h eτ (cid:61) ( V V ) (cid:105) + g Z (cid:48) M πm Z (cid:48) (cid:88) j =2 , (cid:40) M M j ( M + M j ) (cid:104) (cid:61) ( V ∗ V j − V ∗ V j ) (cid:60) ( V ∗ V j − V ∗ V j ) (cid:105) + c α v S M j s √ M + M j )( s − m H ) (cid:61) ( V ∗ V j − V ∗ V j ) (cid:60) ( V ∗ V j − V ∗ V j ) × (cid:104) h eµ (cid:61) ( V V ) + h eτ (cid:61) ( V V ) (cid:105)(cid:41) , (11)where s = 4 M / (1 − v / s α = sin α ( c α = cos α ), and we show only the leading terms in v rel and M /m Z (cid:48) . The vev of S can be replaced by the m Z (cid:48) using v S = m Z (cid:48) /g Z (cid:48) . Near theresonance region, i.e. m H i ≈ M , the propagator, 1 / ( s − m H i ), should be appropriatelyreplaced by the Breit-Wigner form, 1 / ( s − m H i + im H i Γ H i ). The 1st line results from Fig. 1(a), the 2nd line from N contribution of Fig. 1 (b-c), and the 3rd line is the interference termbetween them. The 4th line comes from N , contribution of Fig. 1 (b-c), whose interferenceterm with Fig. 1 (a) is the last term. We assume the mixing angle α in the scalar sectoris small, and we suppressed terms with s α from the 2nd line on. As can be seen clearly in(11), the σv rel has enhancement factor M /m Z (cid:48) compared to naive estimate which is givenby σv rel ∼ g Z (cid:48) /M . For the electroweak scale N and m Z (cid:48) ∼
100 MeV, the enhancementfactor can be of order 10 , which can compensate the suppression due to g Z (cid:48) ∼ − , togive the correct relic density.We scanned the region which can explain muon ( g −
2) anomaly in ( m Z (cid:48) , g Z (cid:48) ) plane [4],which can also be seen in the right panel of Fig. 3. For other parameters, we set α = 10 − ,m H = 125 GeV ,λ = λ = λ = 1 ,m η ± = m η R = m η I = 10 TeV , (12)where m η ± and m η R ( I ) are charged- and neutral-masses from inert scalar doublet η . Thechange of the above parameters does not change our results much. And we scanned in therange 0 < m H < √ πm Z (cid:48) /g Z (cid:48)
10 GeV < M ee , M µτ <
100 GeV − π < h eµ , h eτ < π − π < θ R < π, (13)where used m H ≈ √ λ v S to set the maximum value of m H . With this scan, we get M (cid:46)
100 GeV and M (cid:46) M (left panel) and M (right panel). The horizontallines represent ± σ values of Planck result, Ω h = 0 . ± . M (cid:38) t − channel N contribution which isnot suppressed by v can be important if it is not too heavy. The neutrino masses are sensitive to Yukawa couplings f i ( i = e, µ, τ ) in (3) and are not strongly correlatedwith the dark matter phenomenology FIG. 2: The relic density versus M (left panel) and M (right panel). The horizontal lines represent ± σ values of Planck result, Ω h = 0 . ± . M , M ) plane (left panel) and ( m Z (cid:48) , g Z (cid:48) ) plane (right panel). All thepoints can explain the ( g − µ at 2 σ level. The green points satisfy 0 . < Ω h < .
14, the bluepoints Ω h < .
1, and the gray points Ω h > .
14. In the right panel the straight lines correspondto m Z (cid:48) /g Z (cid:48) = 100 , ,
300 GeV from the left.
In Fig. 3, we show scatter plots in ( M , M ) plane (left panel) and ( m Z (cid:48) , g Z (cid:48) ) plane(right panel). All the points can explain the ( g − µ at 2 σ level. The green points satisfy0 . < Ω h < .
14, the blue points Ω h < .
1, and the Gray points Ω h > .
14. In the rightpanel the straight lines correspond to M Z (cid:48) /g Z (cid:48) = 100 , ,
300 GeV from the left. We cansee that the relic abundance of our universe can be explained if N is not too light ( i.e. if M (cid:38) N has electroweak scale mass. The right panel shows that the correctrelic density can be obtained if Z (cid:48) is not too light. If Z (cid:48) is too light, i.e. m Z (cid:48) (cid:46)
40 MeV, theannihilation cross section becomes too large and the relic density becomes too small.Since Z (cid:48) does not couple to quarks directly, our model does not have tree-level diagramfor the direct detection of dark matter off nucleons. At one-loop level, Z (cid:48) can mix withphoton via virtual (cid:96) + (cid:96) − ( (cid:96) = µ, τ ) pair production and annihilation diagrams. Through thismixing the dark matter can scatter off nucleons. To estimate the elastic scattering crosssection for direct detection it is convenient to introduce effective operator [12] L eff = 1Λ ( N γ µ γ N )( (cid:96)γ µ (cid:96) ) , (14)where (cid:96) = µ, τ . The cut-off scale Λ is approximately given by Λ = m Z (cid:48) /g Z (cid:48) . As can be seen inthe right panel of Fig. 3, the cut-off scale is in the electroweak scale. Due to Majorana natureof N , the vector current N γ µ N vanishes identically. The elastic scattering, however, is p − wave and the cross section is suppressed by v ≈ − [12].If we did not consider the muon ( g − U (1) µ − τ gauge boson is also viable in theheavier m Z (cid:48) or larger g Z (cid:48) parameter region. In this case the Z (cid:48) can be searched for atcolliders through 4 µ , 2 µ τ , 4 τ production processes or missing E T signals in associationwith 2 µ or 2 τ events [10]. The parameter region with m Z (cid:48) ∼ O (10) GeV and g Z (cid:48) (cid:38) . Z → µ [14, 15]. In the on-going LHC Run IIexperiment wider region of parameter space will be covered [16]. The region of our interest, i.e. , g Z (cid:48) ∼ O (10 − ) and m Z (cid:48) ∼ O (100) MeV, may be searched for with dedicated study ofspecific topology of events including the one such as lepton jet [4]. This low m Z (cid:48) would betested better at future high luminosity colliders such as FCC at CERN, Belle II, or plannedneutrino facility LBNE.The large ν µ flux from the dark matter annihilation at the galactic center can also be asignal of our model [10]. Those neutrinos can give additional contributions to the upward-going muon signals at the Super-Kamiokande. Although the photons emitted from the muonscould contribute to the gamma rays from the galactic center, the cross section turns out tobe too small to explain the possible excess of gamma ray events from the Fermi-LAT [17]. IV. CONCLUSIONS
In this paper we considered dark matter phenomenology of right-handed neutrino darkmatter candidate in an extension of Ma’s scotogenic model with U (1) µ − τ gauge symmetry.We showed that we can explain the correct relic density of dark matter and the anomaly0of muon ( g −
2) at the same time. We need light Z (cid:48) ( m Z (cid:48) (cid:46)
400 MeV) and small U (1) µ − τ gauge coupling (3 × − (cid:46) g Z (cid:48) (cid:46) − ). Although the gauge coupling constant is small weshowed that the longitudinal polarization of Z (cid:48) gauge boson in N N → Z (cid:48) Z (cid:48) annihilationprocess can give large enhancement factor M /m Z (cid:48) to get the correct relic abundance ofdark matter. Our model is not strongly constrained by the direct detection experiments ofdark matter. However, the Z (cid:48) gauge boson can be searched for at the current LHC Run IIand future high luminosity hadron or neutrino collider experiments. Acknowledgments
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