Gauged U(1)_{L_{\mu}-L_{\tau}} model in light of muon g-2 anomaly, neutrino mass and dark matter phenomenology

# Gauged U(1)Lμ−Lτ model in light of muon g−2 anomaly, neutrino mass and dark matter phenomenology

Sudhanwa Patra Center of Excellence in Theoretical and Mathematical Sciences, Siksha ’O’ Anusandhan University, Bhubaneswar-751030, Odisha, India    Soumya Rao ARC Centre of Excellence for Particle Physics at the Terascale, Department of Physics,
Nirakar Sahoo Department of Physics, Indian Institute of Technology Hyderabad, Kandi, Sangareddy, 502285, Medak, Telengana, India    Narendra Sahu Department of Physics, Indian Institute of Technology Hyderabad, Kandi, Sangareddy, 502285, Medak, Telengana, India
###### Abstract

Gauged model has been advocated for a long time in light of muon anomaly, which is a more than discrepancy between the experimental measurement and the standard model prediction. We augment this model with three right-handed neutrinos and a vector-like singlet fermion to explain simultaneously the non-zero neutrino mass and dark matter content of the Universe, while satisfying anomalous muon constraints. It is shown that in a large parameter space of this model we can explain positron excess, observed at PAMELA, Fermi-LAT and AMS-02, through dark matter annihilation, while satisfying the relic density and direct detection constraints.

## I Introduction

The standard model (SM) of elementary particle physics, which is based on the gauge group is very successful in explaining the fundamental interactions of nature. With the recent discovery of Higgs at LHC, the SM seems to be complete. However, it has certain limitations. For example, the muon anomaly, which is a discrepancy between the observation and SM measurement with more than confidence level Miller:2007kk (). Similarly, it does not explain sub-eV masses of active neutrinos as confirmed by long baseline oscillation experiments Fukuda:2001nk (). Moreover, it does not accommodate any particle candidate of dark matter (DM) whose existence is strongly supported by galaxy rotation curve, gravitational lensing and large scale structure of the universe (DM_review, ). In fact, the DM constitutes about of the total energy budget of the universe as precisely measured by the satellite experiments WMAP wmap () and PLANCK PLANCK ().

At present LHC is the main energy frontier and is trying to probe many aspects of physics beyond the SM. An attractive way of probing new physics is to search for a -gauge boson which will indicate an existence of symmetry. Within the SM, we have accidental global symmetries , where is the baryon number, and , where is the total lepton number. Note that and are anomalous and can not be gauged without adding any ad hoc fermions to the SM. However, the differences between any two lepton flavours, i.e., , with , are anomaly free and can be gauged without any addition of extra fermions to the SM. Among these extensions the most discussed one is the gauged  Baek:2001kca (); Ma:2001md (); Heeck:2011wj (); Heeck:2010ub (); Heeck:2010pg (); Ota:2006xr (); Rodejohann:2005ru (); Xing:2015fdg (); Rivera-Agudelo:2016kjj (); Choubey:2004hn (); He:1991qd (); Araki:2015mya (); Fuyuto:2015gmk (); Heeck:2016xkh (); Altmannshofer:2014cfa (); Crivellin:2015lwa (); Crivellin:2015mga (); Heeck:2014qea (); Shuve:2014doa (); Salvioni:2009jp (); Yin:2009mc (); Bi:2009uj (); Adhikary:2006rf (); Ma:2001tb (); Bell:2000vh (); He:1994aq (); Kim:2015fpa (); Harigaya:2013twa (); Fuki:2006xw (); Elahi:2015vzh (); Altmannshofer:2016oaq ()The interactions of corresponding gauge boson are restricted to only and families of leptons and therefore it significantly contribute to muon anomaly, which is a discrepancy between the observation and SM measurement with more than confidence level. Moreover, does not have any coupling with the electron family. Therefore, it can easily avoid the LEP bound: TeV Carena:2004xs (). So, in this scenario a - mass can vary from a few MeV to TeV which can in principle be probed at LHC and at future energy frontiers.

In this paper we revisit the gauged model in light of muon anomaly, neutrino mass and DM phenomenology. We augment the SM by including three right handed neutrinos: , and , which are singlets under the SM gauge group, and a vector like colorless neutral fermion . We also add an extra SM singlet scalar . All these particles except , are charged under , though singlet under the SM gauge group. When acquires a vacuum expectation value (vev), the breaks to a remnant symmetry under which is odd while all other particles are even. As a result serves as a candidate of DM. The smallness of neutrino mass is also explained in a type-I see-saw framework with the presence of right handed neutrinos , and whose masses are generated from the vev of scalar field .

In this model the relic abundance of DM () is obtained via its annihilation to muon and tauon family of leptons through the exchange of gauge boson . We show that the relic density crucially depends on gauge boson mass and its coupling . In particular, we find that the observed relic density requires for MeV. However, if then we get an over abundance of DM, while these couplings are compatible with the observed muon anomaly. We resolve this conflict by adding an extra singlet scalar doubly charged under , which can drain out the large DM abundance via the annihilation process: . As a result, the parameter space of the model satisfying muon anomaly can be reconciled with the observed relic abundance of DM. We further show that the acceptable region of parameter space for observed relic density and muon anomaly is strongly constrained by null detection of DM at Xenon-100 Aprile:2012nq () and LUX Akerib:2013tjd (). Moreover, the compatibility of the present framework with indirect detection signals of DM is also checked. In particular, we confront the acceptable parameter space with the latest positron data from PAMELA Adriani:2008zr (); Adriani:2010ib (), Fermi-LAT FermiLAT:2011ab () and AMS-02 Aguilar:2013qda (); Accardo:2014lma ().

The paper is arranged as follows. In section-II, we describe in details the different aspects of the model. Section-III is devoted to show the allowed parameter space from muon anomaly. In section-IV, we estimate the neutrino mass within the allowed parameter space. Section V, VI and VII are devoted to obtain constraints on model parameters from the relic density, direct and indirect search of DM. In section-VIII, we lay the conclusions with some outlook.

## Ii The model for muon g−2 anomaly, neutrino mass and dark matter

We consider the gauge extension of the SM with extra symmetry (from now on referred to as “gauged model”) where difference between muon and tau lepton numbers is defined as a local gauge symmetry Baek:2001kca (); Ma:2001md (); Heeck:2011wj (); Heeck:2010ub (); Heeck:2010pg (); Ota:2006xr (); Rodejohann:2005ru (); Xing:2015fdg (); Rivera-Agudelo:2016kjj (); Choubey:2004hn (); He:1991qd (); Elahi:2015vzh (); Araki:2015mya (); Fuyuto:2015gmk (); Heeck:2016xkh (); Altmannshofer:2014cfa (); Altmannshofer:2016oaq (); Crivellin:2015lwa (); Crivellin:2015mga (); Heeck:2014qea (); Shuve:2014doa (); Salvioni:2009jp (); Yin:2009mc (); Bi:2009uj (); Adhikary:2006rf (); Ma:2001tb (); Bell:2000vh (); He:1994aq (); Kim:2015fpa (); Harigaya:2013twa (); Fuki:2006xw (). The advantage of considering the gauged model is that the theory is free from any gauge anomaly without introduction of additional fermions. We break the gauge symmetry to a residual discrete symmetry and explore the possibility of having non-zero neutrino mass and a viable candidate of DM.

### ii.1 Spontaneous symmetry breaking

The spontaneous symmetry breaking of gauged model is given by:

 GLμ−Lτ⟨S⟩,⟨η⟩⟶GSM×Z2⟨H⟩⟶SU(3)C×U(1)em×Z2, (1)

where

 GLμ−Lτ≡SU(3)C×SU(2)L×U(1)Y×U(1)Lμ−Lτ, GSM≡SU(3)C×SU(2)L×U(1)Y

At first, the spontaneous symmetry breaking of is achieved by assigning non-zero vacuum expectation values (vevs) to complex scalar field and . The subsequent stage of symmetry breaking is obtained with the SM Higgs providing masses to known charged fermions. The complete spectrum of the gauged model in light of DM and neutrino mass is provided in Table I where the respective quantum numbers are presented under . To the usual quarks and leptons, we have introduced additional neutral fermions for light neutrino mass generation via seesaw mechanism and a vector like Dirac fermion for the candidate of DM, being odd under the residual discrete symmetry . we note that except all other particles are even under the symmetry.

### ii.2 Interaction Lagrangian

The complete interaction Lagrangian for the gauged model is given by

 LLμ−Lτ =i¯¯¯¯¯Ne⧸∂Ne+i¯¯¯¯¯Nμ(⧸∂+igμτZ′μγμ)Nμ+i¯¯¯¯¯Nτ(⧸∂−igμτZ′μγμ)Nτ −gμτ(¯¯¯μγμμ+¯¯¯¯¯νμγμPLνμ−¯¯¯τγμτ−¯¯¯¯¯ντγμPLντ)Z′μ −Mee¯¯¯¯¯¯¯NceNe−(λeμS⋆¯¯¯¯¯¯¯NceNμ+h.c)−(λeτS¯¯¯¯¯¯¯NceNτ+h.c) −(λμμη⋆¯¯¯¯¯¯¯NcμNμ+h.c)−(λττη¯¯¯¯¯¯¯NcτNτ+h.c) −(Yee¯¯¯¯¯¯Le~HNe+Yμμ¯¯¯¯¯¯Lμ~HNμ+Yττ¯¯¯¯¯¯Lτ~HNτ+h.c) +i¯¯¯¯χ(⧸∂+igμτZ′μγμ)χ−Mχ¯¯¯¯χχ−fχ¯¯¯¯¯χcχη⋆ −14FμνZ′FZ′μν+ϵ4FμνZ′Fμν +|(∂μ+igμτZ′μ)S|2−μ2SS†S+λS(S†S)2+|(∂μ+i2gμτZ′μ)η|2−μ2ηη†η+λη(η†η)2 +λHS(H†H)(S†S)+λHη(H†H)(η†η)+ληS(η†η)(S†S)+μηSSSη⋆+LSM, (2)

Here is the SM Lagrangian. We denote here as the new gauge boson for and the corresponding field strength tensor as . The gauge coupling corresponding to is defined as (as mentioned in section I).

### ii.3 Scalar masses and mixing

The scalar potential of the model is given by

 V(H,S) =−μ2HH†H+λH(H†H)2−μ2ηη†η+λη(η†η)2−μ2SS†S+λS(S†S)2 +λSH(H†H)(S†S)+λHη(H†H)(η†η)+ληS(η†η)(S†S)+μηSSSη⋆ (3)

where is the SM Higgs doublet and , are the complex scalar singlets under SM, while charged under . The neutral complex scalars , and can be parametrised as follows:

 H0=1√2(vH+h)+i√2G0, S=1√2(vS+s)+i√2A, η=1√2(vη+η)+i√2B,

The mass matrix for the neutral scalars is given by

 M2=⎛⎜ ⎜⎝2λHv2HλSHvHvSλHηvHvηλSHvHvS2λSv2SληSvSvη+μηSvSλHηvHvηληSvSvη+μηSvS2ληv2η⎞⎟ ⎟⎠ (5)

This is a symmetric mass matrix. So it can be diagonalised by a unitary matrix:

 V†M2V=Diagonal(M2h,M2S,M2η) (6)

We identify as the physical mass of the SM Higgs, while and are the masses of additional scalars and respectively. Since and are singlets, their masses can vary from sub-GeV to TeV region. For a typical set of values: , the physical masses are found to be GeV, GeV, GeV and the mixing between and field is . We will study the importance of field while calculating the relic abundance of DM. The mixing between and field is required to be small as it plays a dominant role in the direct detection of DM. We will show in Fig.5 that if the mixing angle is large then it will kill almost all the relic abundance parameter space.

### ii.4 Mixing in the Gauge Sector

The breaking of gauged symmetry by the vev of and gives rise to a massive neutral gauge boson which couples to only muon and tauon families of leptons. In the tree level there is no mixing between the SM gauge boson and . However at one loop level, there is a mixing between and through the exchange of muon and tauon families of leptons as shown in the figure 1.

The loop factor can be estimated as

 Πμν(q2)=−(q2gμν−qμqν)13116π2(gμτCVg2cosθW)Log(m2fΛ2) (7)

where is the Weinberg angle, is the vector coupling of SM fermions with boson, is the cut off scale of the theory and is the mass of the charged fermion running in the loop. In the gauge basis, the mass matrix is given by

 M22=(M2Z0ΠΠ~M2Z′) (8)

where is given by and GeV. Thus the mixing angle is given by

 tan2θZ=2Π~M2Z′−M2Z0 (9)

Diagonalising the mass matrix (8) we get the eigen values:

 M2Z=M2Z0−M2Z′sin2θZcos2θZ M2Z′=~M2Z′−M2Zsin2θZcos2θZ (10)

where and are the physical masses of and gauge bosons. The mixing angle has to chosen in such a way that the physical mass of Z-boson should be obtained within the current uncertainty of the SM boson mass pdg (). It can be computed from equation (II.4) as follows:

 MZ−MZ0MZ0=M2Z0−M2Z′2M2Z0tan2θZ≤4.6×10−5 (11)

For we get .

## Iii Muon g−2 Anomaly

The magnetic moment of muon is given by

 −→μμ=gμ(q2m)→S, (12)

where is the gyromagnetic ratio and its value is for a structureless, spin particle of mass and charge . Any radiative correction, which couples to the muon spin to the virtual fields, contributes to its magnetic moment and is given by

 aμ=12(gμ−2) (13)

At present there is a more than discrepancy between the experimental measurement Bennett:2006fi () and the SM prediction Miller:2007kk () of value. This is given by:

 Δaμ=aexptμ−aSMμ=(295±88)×10−11. (14)

In the present model, the new gauge boson contributes to and is given by Baek:2008nz ()

 Δaμ=α′2π∫10dx2m2μx2(1−x)x2m2μ+(1−x)M2Z′≈α′2π2m2μ3M2Z′, (15)

where . The above equation implies that the discrepancy between the experimental measurement Bennett:2006fi () and the SM prediction Miller:2007kk () of value can be explained in a large region of parameter space as shown in Fig. (3).

## Iv Neutrino mass

In order to account for tiny non-zero neutrino masses for light neutrinos, we extend the minimal gauged model with additional neutral fermions where the quantum numbers in the parentheses are the charge. The relevant Yukawa interaction terms are given by

 L= −12Mee¯¯¯¯¯¯¯NceNe−12Mμτ¯¯¯¯¯¯¯NcμNτ−(λeμS⋆¯¯¯¯¯¯¯NceNμ+h.c)−(λeτS¯¯¯¯¯¯¯NceNτ+h.c) −(λμμη⋆¯¯¯¯¯¯¯NcμNμ+h.c)−(λττη¯¯¯¯¯¯¯NcτNτ+h.c) −(Yee¯¯¯¯¯¯Le~HNe+Yμμ¯¯¯¯¯¯Lμ~HNμ+Yττ¯¯¯¯¯¯Lτ~HNτ+h.c) = −12NTαC−1MRαβNβ+MDαβ¯¯¯¯¯ναNβ+h.c. (16)

where the Dirac and Majorana neutrino mass matrices are given by

 MR=⎛⎜⎝MeeλeμvSλeτvSλeμvSλμμvηMμτλeτvSMμτλττvη⎞⎟⎠,MD=⎛⎜⎝YeevH000YμμvH000YττvH⎞⎟⎠, (17)

Using seesaw approximation, the light neutrino mass matrix can be read as

 mν≃−MDM−1RMTD. (18)

We illustrate here a specific scenario where not only the resulting Dirac neutrino mass matrix is diagonal but also degenerate. As a result, we can express . One can express heavy Majorana neutrino mass matrix in terms of light neutrino mass matrix as

 MR=m2dm−1ν. (19)

Thus, one can reconstruct using neutrino oscillation parameters and GeV. As we know that light neutrino mass matrix is diagonalised by the PMNS mixing matrix as

 mdiag.ν=U†PMNSmνU∗PMNS=diag.{m1,m2,m2}

where are the light neutrino mass eigenvalues. The PMNS mixing matrix is generally parametrized as

 UPMNS=⎛⎜⎝c12c13s12c13s13e−iδ−s12c23−c12s13s23eiδc12c23−s12s13s23eiδc13s23s12s23−c12s13c23eiδ−c12s23−s12s13c23eiδc13c23⎞⎟⎠⋅P (20)

where , (for ), and . Here we denoted Dirac phase as and Majorana phases as .

For a numerical example, we consider the best-fit values of the oscillation parameters, the atmospheric mixing angle , solar angle , the reactor mixing angle , and the Dirac CP phase (Majorana phases assumed to be zero here for simplicity i.e, ). The PMNS mixing matrix for this best-fit oscillation parameters is estimated to be

 UPMNS=⎛⎜⎝0.81680.5552−0.1265−0.0919i−0.3461−0.0510i0.6604−0.0347i0.66340.4551−0.0563i−0.5028−0.0382i0.7316⎞⎟⎠. (21)

We also use the best-fit values of mass squared differences and . As we do not know the sign of , the pattern of light neutrinos could be normal hierarchy (NH) with ,

 m2=√m21+Δm2s,m3=√m21+Δm2a,

or, the inverted hierarchy (IH) with ,

 m1=√m23+Δm2a,m2=√m23+Δm2a+Δm2s..

Now, one can use these oscillation parameters and GeV, the mass matrix for heavy neutrinos is expressed as

 MR=10−8GeV2(UPMNSmdiag.νUTPMNS)−1. (22)

Using eV, the masses for heavy neutrinos are found to be GeV, GeV and GeV. The same algebra can be extended for inverted hierarchy and quasi-degenerate pattern of light neutrinos for deriving structure of .

## V Relic abundance of dark matter

The local is broken to a remnant symmetry under which is odd and all other fields are even. As a result becomes a viable candidate for DM. We explore the parameter space allowed by relic abundance and null detection of DM at direct search experiments in the following two cases:
(a) in absence of
(b) in presence of .

### v.1 Relic abundance in absence of η

For simplicity, we assume that the right handed neutrinos and as well as the scalar field are heavier than the mass. Now in absence of  111In absence of the neutrino mass will not be affected., the relevant diagrams that contribute to the relic abundance of DM are shown in Fig. (2).

Since the null detection of DM at direct search experiments, such as Xenon-100 and LUX restricts the mixing to be small (), the dominant contribution to relic abundance, below the threshold of , comes from the s-channel annihilation: through the exchange of . Due to the resonance effect this cross-section dominates. We have shown in Fig. 3, the correct relic abundance of DM in the plane of and . Below the red line the annihilation cross-section through exchange is small due to small gauge coupling and therefore, we always get an over abundance of DM. The constraints from muon anomaly and direct detection of DM via mixing are also shown in the same plot for comparison. From Fig. 3, we see that in a large parameter space we can not get any point which satisfies both relic abundance of DM as well as muon anomaly constraints. Therefore, it does not serve our purpose. We resolve the above mentioned issues in presence of the scalar field , where the number inside the parenthesis is the charge under .

### v.2 Relic abundance in presence of η

In presence of the SM singlet scalar field , the new annihilation channels , shown in Fig. 4, and open up in addition to the earlier mentioned channels, shown in fig. (2). However, in the region of small gauge coupling , the dominant channel for the relic abundance of DM is . The other channel: is suppressed due to the small mixing angle , required by null detection of DM at direct search experiments. We assume that the mass of to be less than a GeV as discussed in section II.3. In this case the analytic expression for the cross-section of is given by:

 ⟨σ|v⟩(χχ→ηη)=1128π1(1−M2η2M2χ)2f4χM2χ(1−M2ηM2χ)3/2 (23)

From the above expression we observe that the cross-section goes as for . Fixing GeV and varying the DM mass and the coupling , we have shown in Fig. 5 the allowed region in the plane of and for the correct relic abundance. The green points show the value of analytic approximation 23, while the red points reveal the result from full calculation using micrOMEGAs micromegas (). The matching of both points indicates that the contribution to relic abundance is solely coming from the channel. From the Fig. 5 it is clear that as the ratio decreases, i.e., increases for a fixed value of , we need a large coupling to get the correct relic abundance. For comparison, we also show the DM-nucleon spin independent elastic cross-section: mediated through the mixing, in the same plot. We see that the allowed mixing angle by LUX data is quite small.

## Vi Direct Detection

We constrain the model parameters from null detection of DM at direct search experiments such as Xenon-100 Aprile:2012nq () and LUX Akerib:2013tjd () in the following two cases:
a. In the absence of
b. In the presence of .
We show that in absence of field, the elastic scattering of DM with nucleon through mixing give stringent constraint on the model parameters: and , as depicted in fig. (3). On the other hand, in the presence of field , the elastic scattering will be possible through the mixing, while inelastic scattering with nucleon will be possible via mixing. In the following we discuss in details the possible constraints on model parameters.

### vi.1 Direct Detection in absence of η

While the direct detection of DM through its elastic scattering with nuclei is a very challenging task, the splendid current sensitivity of present direct DM detection experiments might allow to set stringent limits on parameters of the model, or hopefully enable the observation of signals in near future. In absence of field, the elastic scattering between singlet fermion DM with nuclei is displayed in Fig. 6.

The spin independent DM-nucleon cross-section mediated via the loop induced mixing is given by Goodman:1984dc (); Essig:2007az ()

 σZSI=164πA2μ2rtan2θZGF2√2g2μτM2Z′[Zfpfn+(A−Z)]2f2n, (24)

where is the mass number of the target nucleus, is the reduced mass, is the mass of nucleon (proton or neutron) and and are the interaction strengths (including hadronic uncertainties) of DM with proton and neutron respectively. Here is the atomic number of the target nucleus.

For simplicity we assume conservation of isospin, i.e. . The value of is varied within the range: hadronic_uncertainty (). If we take , the central value, then from Eqn. (24) we get the total cross-section per nucleon to be

 σZSI≃7.6×10−46cm2tan2θZg2μτM2Z′. (25)

for the DM mass of GeV.

Since the -boson mass puts a stringent constraint on the mixing parameter to be  Hook:2010tw (); Babu:1997st (), we choose the maximum allowed value () and plot the smallest spin independent direct DM detection cross-section (), allowed by LUX, in the plane of versus as shown in fig (3). The plot follows a straight line as expected from equation (24) and shown by the green line in fig. (3). Any values above that line will be allowed by the LUX limit.

### vi.2 Direct Detection in presence of η

In presence of the field both elastic and inelastic scattering between DM and the nuclei is possible. The elastic scattering is mediated through mixing while inelastic scattering is mediated by the mixing.

#### vi.2.1 Elastic scattering of dark matter

The spin-independent scattering of DM with nuclei is a t-channel exchange diagram as shown in Fig. 7 through the mixing of scalar singlet with the SM Higgs . The elastic scattering cross section off a nucleon is given byGoodman:1984dc (); Essig:2007az () :

 σηhSI=μ2rπA2[Zfp+(A−Z)fn]2 (26)

where is the reduced mass, Z and A are respectively atomic and mass number of the target nucleus. In the above equation and are the effective interaction strengths of the DM with the proton and neutron of the target nucleus and are given by:

 fp,n=∑q=u,d,sfp,nTqαqmp,nmq+227fp,nTG∑q=c,t,bαqmp,nmq (27)

with

 αq=fχsinθηhM2η(mqvH) (28)

In Eq. (27), the different coupling strengths between the DM and light quarks are given by DM_review () , ,, ,,. The coupling of DM with the gluons in target nuclei is parameterised by

 fnTG=1−∑q=u,,d,sfnTq. (29)

Thus from Eqs. 26, 27, 28, 29, the spin-independent DM-nucleon interaction through mixing is given by:

 σηhSI = μ2rf2χsin2θηhπA2M4η (30) × [ZmpvH(fpTu+fpTd+fpTs+29