Muon anomalous magnetic moment and positron excess at AMS-02 in a gauged horizontal symmetric model

We studied an extension of the standard model with a fourth generation of fermions to explain the discrepancy in the muon $(g-2)$ and explain the positron excess seen in the AMS-02 experiment. We introduce a gauged $SU(2)_{HV}$ horizontal symmetry between the muon and the 4th generation lepton families. The 4th generation right-handed neutrino is identified as the dark matter with mass $\sim 700$ GeV. The dark matter annihilates only to $(\mu^+ \mu^-)$ and $(\nu^c_\mu ~\nu_\mu)$ states via $SU(2)_{HV}$ gauge boson. The $SU(2)_{HV}$ gauge boson with mass $\sim 1.4$ TeV gives an adequate contribution to the $(g-2)$ of muon and fulfill the experimental constraint from BNL measurement. The higgs production constraints from 4th generation fermions is evaded by extending the higgs sector.

where a µ is the anomalous magnetic moment in the unit of e/2m µ . In the standard model, contribution of weak interaction to muon magnetic magnetic dipole moment (MDM) goes as a W µ ∝ m 2 µ /M 2 W and we have a SM µ = 19.48 × 10 −10 [7]. In minimal supersymmetric standard model (MSSM) [8,9], we get contributions to muon MDM from neutralino-smuon and chargino-sneutrino loops. In all MSSM diagrams there still a m µ suppression in g − 2 arising from the following cases: (a) In case of bino in the loop, the mixing between the left and right handed smuons is ∝ m µ tanβ (b) In case of winohiggsino or bino-higgsino in the loop, the higgsino coupling with smuon is ∝ y µ , so there is a m µ suppression (c) In the case of chargino-sneutrino in the loop, the higgsino-muon coupling is ∝ y µ , which again gives rise to m µ suppression. Therefor in MSSM a MSSM µ ∝ m 2 µ /M 2 SUSY , where M SUSY is proportional to the mass of the SUSY particles in the loop.
One can evade the muon mass suppression in g − 2 with a horizontal gauge symmetry. In [10] a horizontal U(1) Lµ−Lτ symmetry was used in which muon g − 2 is proportional to m τ and a µ ∝ m µ m τ /m 2 Z , where L µ − L τ gauge boson mass m Z ∝ 100 GeV gives the required a µ . Models with m Z ≤ 1 TeV are ruled out at LHC.
Recently there have been model independent analysis of the beyond SM particles which can give a contribution to a µ [11]. The new physics needed to explain muon g − 2 has also been related to dark matter [12,13] and the implication of this new physics in LHC searches has been studied [14].
In this paper we relate the anomaly of muon g − 2 to the excess of positron signal which extend upto 350 GeV observed by AMS-02 experiment [15]. An analysis of AMS-02 data suggest that a dark matter annihilation interpretation would imply that the annihilation final states are either µ or τ [17,18]. Dark matter annihilation into e ± pairs would give a peak in positron signal, which is not seen in the positron spectrum . The AMS-02 experiment does not observe an excess, beyond the cosmic ray background, in the antiproton flux [24].
In this paper we introduced a 4th generation vector lepton doublet and a SU(2) H gauge symmetry between the muon family and 4th generation vector lepton doublet. In this model ∆a µ generated by the charged 4th generation lepton and the SU(2) H gauge boson in the loop is, By taking the 4th generation lepton doublet mass, m L − 4 ∼ 1 TeV, and the SU(2) H gauge boson mass M V + ∼ 2 TeV, we get the required contribution to ∆a µ (Eq.1).
In this model the neutral components of L 4 called L 0 4L,4R act as dark matter (DM). The annihilation through the SU(2) H gauge boson V 3 produces µ final state, which is used to explain AMS-02 [15] positron data by taking the dark matter mass m χ 1 ∼ 1 TeV. Assuming the V 3 mass to be M V 3 ≈ 2m χ 1 , we get a resonant enhancement [21][22][23] in the annihilation cross-section (CS) that is used to get correct relic density Ωh 2 = 0.1199 ± 0.0027 [25,26].
To satisfy the AMS-02 data [15], we require a boost factor of O ∼ (100) [18][19][20], that comes from Sommerfeld enhancement [27] in our model. This paper is organized as follows: In Sec.2, we describe the model and dark matter and neutrino mass arising. In Sec.3 we discuss the fit to AMS-02 data. In Sec.4, we study the one-loop diagram which contribute to muon g − 2 in the model. We give our conclusion in Sec.5.

MODEL
In addition to the weak SU(2) W doublets L L = (ν i , E i ) T L , Q L = (U j , D j ) T L , and SU(2) W singlets E iR , U jR , D jR , where (i = e, µ, τ ) and (j = u, d, c, s, t, b), we introduce two extra SU(2) W doublets, with Y = −1, which have a vector coupling with the SM gauge bosons W µ and similarly for Z µ boson (here g denotes the SU(2) W weak coupling). The vector coupling of the 4th generation SU(2) W lepton doublets ensures anomaly cancellation without introducing 4th generation of quarks doublets. In addition to the vector-like lepton generation, we also add three right handed neutrinos (ν iR ) in the model.
We introduce a SU(2) H horizontal gauge symmetry between (L 4L , L 4R ) and the muon family L µ = (ν µ , µ − ) L , µ R and ν µR , which can be organized in the form of SU(2) H doublets as follows, These doublets couple to the SU(2) H horizontal symmetry gauge bosons V ± µ = (V 1 µ ∓iV 2 µ )/ √ 2 and V 3µ as, where τ is the 2 × 2 matrix representation of SU(2) H , α is the SU(2) H family index (Eq.5), and g H represents the SU(2) H gauge coupling. Here V ± µ is the SU(2) H charge changing boson, which connect the L µ and the L 4 families and V 3µ is the "neutral" current gauge boson of SU(2) H . To give masses to the SU(2) H gauge bosons triplet V a µ , we introduce a SU(2) H scalar doublet ξ, which has no SU(2) H invariant Yukawa couplings with the SM and the 4th generation leptons.
In our model we also add a scalar triplet ∆ with Y = 2, which generate the Majorana mass for the neutrinos and 4th generation neutral leptons, that are the dark matter in this model.
The scalar fields in the model including the SM higgs doublet φ can be parametrized as, Here  Table (I), and SU(2) H quantum numbers are given in Table (II).  The gauge couplings of the scalar fields φ, ξ, ∆ and η are given by the Lagrangian: The masses of the gauge bosons generated after the corresponding scalars take their vevs are, and so the electroweak ρ parameter gets modified as, we take the vev of the triplet higgs ∆, v ∆ ∼ O(1)GeV due to the requirement that the electroweak ρ parameter, ρ = 1.0004 +0.0029 −0.0011 [7]. The renormalizable potential of the scalars field named V (φ, ξ, ∆, η) which is invariant under In our model, the parameters in the potential are chosen such that the vevs of the scalars for phenomenological reasons, which we will discuss in the following sections.

Dark matter mass
In our model we treat the neutral components L 0 4L,4R of L 4 as the dark matter. There are both Dirac as well as Majorana mass terms possible for (L 0 4L , L 0 4R ) and given by the Lagrangian, where y χ D and y χ (L,R) are the Yukawa couplings of vector-like leptons with η and ∆ respectively. After the scalars η and ∆ take their vevs, they generate Dirac mass m χ D = y χ D v η and The Dirac mass term similar to Eq.(12) also generate the mass for charged lepton L − 4 . In the (L 0 4L , L 0 4R ) basis, the mass matrix of L 0 4 is, The mass eigenstates are pseudo-Dirac with eigenvalues, with δ χ = (M χ M L + M χ M R )/2, where δ χ denotes the mass splitting between the states χ 1 and χ 2 . The corresponding mass eigenstates are, The coupling of vector doublets to the Z boson is given as, We assume that δ χ ∼ 100 KeV so that the Z mediated scattering between the dark matter χ 1 and nuclei is suppressed to evade the bounds from direct detection experiments [28,29].

Neutrino mass
In our model we introduced right-handed neutrinos that can mix with the neutral component of the vector-like lepton. In the presence of ν iR (i = e, µ, τ ), neutrinos will contain a Dirac mass in addition to the Majorana mass and their masses will generate by Type-II see-saw mechanism. The Lagrangian that gives mass to the neutrinos is given by, where y ν D , y ν4 and y ν (L,R) are the Yukawa couplings of leptons to the scalars φ, η and ∆ respectively. After the scalars take their vevs, they generate Dirac mass m ν D = y ν D v φ and Majorana mass M ν (L,R) = y ν (L,R) v η . In the basis (ν L , ν R ), the 6 × 6 mass matrix of ν is, where m ν D and M ν R are the 3 × 3 Dirac and Majorana matrices respectively. After diagonalizing the mass matrix M ν , the light neutrino mass is given by [30], where We choose a m ν D texture such that the leading term m T ν D M −1 ν R m ν D = 0 and the neutrino mass is given by the second order term in M −1 νR . In this way one can get realistic neutrino mass from TeV scale vev of η.

DARK MATTER PHENOMENOLOGY
In our model, we identify the 4th generation neutral fermions (L 0 4L,4R ) as the dark matter, and take their masses ∼ 1 TeV, which is required to fit AMS-02 data [15]. The dominant channels of dark matter annihilation into the standard model particles are shown in Fig.(1).
In this scenario for getting the correct relic density, we consider the resonance channel by taking M V 3 ≈ 2m χ 1 . The annihilation CS for relic density comes out O ∼ 10 −26 cm 3 s −1 with the resonant enhancement. We assume the galactic rms velocity v ∼ 10 −3 and get a further enhancement by the Sommerfeld mechanism [27] to get a CS of O ∼ 10 −24 cm 3 s −1 which contributes to AMS-02 [15]. In the following sections we will discuss this scenario in details.

Relic density
The principal annihilation channels for our model are shown in Fig.(1). The combined annihilation cross-section for these channels is, where g H is the horizontal gauge boson coupling to the dark matter, m χ 1 the dark matter mass, M V 3 and Γ V 3 are the mass and the decay width of new gauge boson respectively. The main contributions to the decay width of V 3 come from the decay modes V 3 → µ + µ − , ν µ ν µ and given as, In the non-relativistic limit, s = 4m 2 χ 1 + m 2 χ 1 v 2 + 4m χ 1 δ χ , where δ χ is the mass splitting between the dark matter mass eigenstates (Eq. [13][14]. The thermal average of annihilation rate is given as [21][22][23], where, where x = m χ 1 /T ; K 1 (x), K 2 (x) represent the modified Bessel functions of the second type and g i is the internal degree of freedom of DM particle. Using Eq.(24) and Eq.(25) in Eq. (23), it can be written as, where z = v 2 . We solve the integral for freeze-out (x = 20) condition with the parameters given in Table(III) and found σv ∼ O(10 −26 cm 3 s −1 ) which gives the correct relic density Ωh 2 = 0.1199 ± 0.0027, consistent with Planck [25] and WMAP [26] data as shown in Fig.(2). To satisfy AMS-02 data [15], we need a boost factor of ∼ O(100) [18][19][20]. We use the Somerfeld enhancement [27] for getting the needed boost factor that we will describe in the following section. In a non-relativistic annihilation process there can be a distribution of the initial states wave functions by the exchange of light particles. In the case of χ 1 χ 1 → µ + µ − annihilation, the Z-exchange ladder (Fig.3) gives a Sommerfeld enhancement [27] to the CS. The Schrodinger eq. obeyed by the dark matter doublet ψ = (χ 1 , χ 2 ) is

Sommerfeld Enhancement
where the 2 × 2 matrix potential in the (χ 1 , χ 2 ) basis is, The potential energy term mixes the χ 1 , χ 2 states. The χ 1 χ 1 → µ + µ − annihilation CS is enhanced by the Sommerfeld factor S, where σ 0 is the CS without the Z exchange ladder and S is determined from the solution of the Schrodinger equation and defined as, The S factor of the inelastic Sommerfeld enhancement process has been computed in [27].
The Sommerfeld enhancement depends on the relativistic velocity v ∼ (10 −3 −10 −4 ), Z coupling α z , Z mass M Z , the dark matter mass m χ 1 and mass splitting δ χ . For the parameters of our model (Table.III), the Sommerfeld enhancement factor is given by [27], where away from the resonance θ − = nπ + π 2 δ − 2 v /2µ. The dimensionless parameters are defined as v = v/α z , δ = δ χ /m χ 1 /α z and µ ∼ z = (M Z /m χ 1 )/α z . By using the values of parameters from Table(III), we get v = 0.16, δ = 0.007 and z = 2. We use these values of dimensionless parameter in Eq.(31) and find the Sommerfeld enhancement factor S = 105, which is used as an input for computing the positron flux for comparison with the AMS-02 data [15].

Comparison with AMS-02 and PAMELA data
We use annihilation CS of χ 1 χ 2 → µ + µ − mediated by V 3 at resonance when M V 3 ≈ 2m χ 1 , to get the correct relic density [25,26], and Somerfeld enhancement [27] for getting enhanced CS required for AMS-02 [15]. We use publicly available code PPPC4DMID [33,34] to compute the positron spectrum dN e + dE from the decay of µ + µ − for 1 TeV dark matter. We then use the GALPROP code [35,36] in which we take the annihilation CS (χ 1 χ 1 → µ + µ − ) and the positron spectrum dN e + dE as an input to the source term, where σv f denotes the annihilation-rate, dN f e + dE positron spectrum and ρ denotes the density of dark matter in the Milky Way halo, which is described by NFW profile [37] in our model, The basic parameters of the GALPROP are chosen as in [31,32] for 2D conventional model. In our model, we consider diffusion coefficient D 0 = 2.8 × 10 28 cm 2 s −1 and Alfven speed v A = 30Kms −1 . We choose, z h = 4kpc and r max = 20kpc, which are the half-width and maximum size for 2D galactic model respectively. In our model, the nucleus spectral index breaks at 9 GeV and spectral index above this is γ n 2 = 2.36 and below γ n 1 = 1.82. The normalization flux of electron at 100GeV is 1.25 × 10 −8 cm −2 s −1 sr −1 GeV −1 and for the case of electron, we take breaking point at 4 GeV and its injection spectral index above 4 GeV is γ el 1 = 2.4 and below γ el 0 = 1.6. After solving the propagation equation by taking the source term (Eq.32) and considering the Sommerfeld enhancement [27] of CS, GALPROP [35,36] gives the desired positron flux. In Fig.(4), we plotted the output of GALPROP code and compared it with the observed AMS-02 data [15]. We see that our positron spectrum fits the AMS-02 data [15] very well. We also check, as shown in Fig.(5) that the γ − ray does not exceed the limits from Fermi-LAT [39] experiment. There is no antiproton excess in cosmic rays secondaries observed by PAMELA [38] and AMS-02 [24]. This fits in with the leptophilic DM of our model where only µ + µ − is produced with an enhanced CS. We calculate the magnetic moment operator, which is given by where F 2 (q 2 ) is the magnetic form factor, which arises from the loop. The anomalous magnetic moment is related to F 2 as ∆a µ = F 2 (q 2 = 0). The vertex factor of the amplitude µ(p )Γ µ µ(p) µ shown in the Fig.(6) is, We perform the integration (see the Appendix) and use the Gorden identity to replace and identify the coefficient of the iσ µν q ν as the magnetic form factor. In the limit of M 2 V + >> m 2 L − 4 , we get the anomalous magnetic moment (see the Appendix).
We note that in Eq.(37) the first term is dominant which shows that in the horizontal symmetry case there is a m µ m L − 4 enhancement over the model discussed in the introduction. We used the parameters from Table.(III) and found the muon MDM, in agreement with the experiment [1]. We plot the relic density and muon g − 2 together within 1σ in Fig.(2). This is clear from the plot that there exists a narrow region in which both constraints are satisfied together.

CONCLUSION
We have related two experimental signatures of possible beyond standard model physics in an canonical model involving an extra vector lepton doublet and a SU(2 H ) horizontal symmetry between the muon family and the L 4 family. An extra vector lepton doublet as asymmetric dark matter has been studied in [40,41]. In our work we introduced horizontal gauge symmetry to get a contribution to the muon g − 2 and to explain AMS-02 data. This model requires SU(2) triplet and singlet higgs of TeV scales and these can be probed in forthcoming LHC searches [42]. The charged fermions and scalars can also contribute to the h → γγ loop [43,44] and this can be a potential test of this model.

Appendix: Muon magnetic moment calculation
In our model contribution to the muon magnetic dipole moment comes from Fig.(6).
The vertex factor of the amplitude is given in Eq. (35), which can be solved using standard Feynman parametrization method as follows, where N µ is the numerator in Eq. (35), and denominator is given by where we use the property x + y + z = 1 because of δ-function appearing in Eq.(A1). To solve it further, we introduce a new momentum variable defined as, In terms of new variable, we can write Eq.(A2) as, where Here we use p 2 = p 2 = m 2 µ , which refers to q 2 = 2m 2 µ − 2p · p . In the numerator of Eq.(35) which is, we have to replace k by l − yp − zp and by this replacement, numerator N µ becomes, now we use contraction identities to simplify the numerator and it takes the following form, where we used σ µν = i 2 [γ µ , γ ν ], and Gorden identity (p µ + p µ ) = 2m µ γ µ + iσ µν q ν (A9) to replace (p µ +p µ ). This is a general form of numerator and so we collect only the coefficient of iσ µν from Eq.(A8), which gives contribution to the anomalous magnetic moment of muon.
We use this coefficient, with Eq.(A4) in Eq. (35) and perform the integrations on variables y and z. At the last we get the following vertex factor, , as defined in Eq.(A5). We do the Wick rotation to find the further solution through momentum integration, using the following identity and get the vertex factor We perform the integration in the limit of M 2 V + >> m 2 L − 4 and find the suitable form of vertex factor as follows, after comparing our result with Eq.(34), we get the magnetic form factor, which in the limit of q 2 = 0 gives the muon anomalous magnetic moment,