Minimally extended left-right symmetric model for dark matter with U(1) portal

A minimal extension of the left-right symmetric model for neutrino masses that includes a vector-like singlet fermion dark matter (DM) is presented with the DM connected to the visible sector via a gauged U(1) portal. We discuss the symmetry breaking in this model and calculate the mass and mixings of the extra heavy neutral gauge boson at the TeV scale. The extra gauge boson can decay to both standard model particles as well to dark matter. We calculate the relic density of the singlet fermion dark matter and its direct detection cross section and use these constraints to obtain the allowed parameter range for the new gauge coupling and the dark matter mass.


Introduction
Left-Right Symmetric Model (LRSM), based on the gauge group SU(2) L × SU(2) R × U(1) B−L [1][2][3], was originally proposed to understand the origin of parity violation in the standard model (SM). The fact that the seesaw mechanism for understanding small neutrino masses [4][5][6][7][8] finds a natural home in these models, has made them more interesting for theory as well as experiments. In particular, the fact that the SU(2) R breaking scale is allowed to be in the TeV range by low energy flavor changing neutral current constraints [9,10] has provided a strong motivation to look for signatures of neutrino mass related physics in colliders such as the Large Hadron Collider (LHC) and low energy processes such as neutrinoless double beta-decay. The current LHC limit on M W R is 4.4 TeV [11] from a study of the ℓℓjj final states and an ATLAS bound of 3 TeV from a study of tb final states [12]. There have also been recent speculations that an anomaly in understanding the CP violating parameter ǫ ′ /ǫ in the SM can possibly be resolved in the left-right models with TeV scale W R [13,14].
In this paper, we explore an extension of this model to understand the origin of dark matter (DM). Two key questions are (i) whether the DM particle is naturally stable and (ii) how it is connected to the SM sector. In recent years, an interesting class of models has been proposed where by adding certain fermion or scalar multiplets to the LRSM makes them naturally stable [15,16] due to symmetries already present in the model and provide candidates for dark matter. It is then connected to the SM sector via the W R and Z R bosons. There have been extensive discussions of DM phenomenology in these models [17][18][19][20]. 1 The most minimal of these models has triplet fermions with B − L = 0. The neutral member of this triplet is the dark matter. This triplet model also leads to coupling unification without the need for supersymmetry [22], which is an interesting property. The structure of these models however implies that there must be constraints between the W R mass (M W R ) and the DM mass (M DM ) for the neutral member of the triplet to be the JHEP12(2018)009 lightest and hence be a viable dark matter. In terms of particle content, the model has six new fermionic states in addition to the usual particle content of the LRSM.
In this paper, we present a slightly more minimal (in the sense of particle content) alternative extension of the left-right model, which also provides a dark matter fermion. The model is based on an extended gauge group SU(2) L × SU(2) R × U(1) B−L × U(1) X with a left-right singlet fermion (called ζ L,R ) being a non-singlet under the extra U(1) X as well as under U(1) B−L in such a way that it is electrically neutral. The extra U(1) X provides a gauge portal to the SM fermions with interesting implications. The detailed phenomenology of dark matter is also very different from the above class of models.
The electric charge formula for this model is The new gauge bosons in the model are W R , Z R and the third neutral vector boson, that we call X, will provide the U(1) portal to dark matter in our model. 2 In this paper, we will assume that all these bosons are in the multi-TeV mass range. An interesting implication of this model is that the extra neutral gauge boson, X, could be lighter than the current LHC bounds for the sequential Z ′ boson [29,30] with mass closer to or even below a TeV. We find two parameter ranges for the masses of the DM fermion ζ and the X boson where we can avoid the stringent bounds on the elastic scattering DM cross section with nuclei by the direct DM detection experiments, in particular, the XENON1T experiment [31]. The paper is organized as follows: in section 2, we present the model content of SU(2) L × SU(2) R × U(1) B−L × U(1) X and the Higgs sector. Section 3 is dedicated to the sector of gauge bosons to obtain their masses and the charged and neutral current interactions, after the spontaneous symmetry breaking (SSB) of the gauge symmetry. In section 4, we calculate the DM relic abundance and identity the model parameter region to reproduce the observed relic density. The constraint from the current direct DM detection experiments, in particular, the XENON1T experiment is derived in section 5. Finally, our concluding comments are cast in section 6.

The model content and the Higgs sector
Our model is based on the gauge group G LR ≡ SU(3) c ×SU(2) L ×SU(2) R ×U(1) B−L ×U(1) X . The quarks and leptons are assigned to the following irreducible representations under G LR :

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where i = 1, 2, 3 is the generation index. The U(1) X charges are zero for all these fermions and are not shown in the above equations. To this fermion content, we add a SU(2) L × SU(2) R singlet vectorlike fermions ζ L and ζ R which have U(1) B−L × U(1) X charges of (+a, −a). In the minimal version of left-right model, the Higgs sector consists of the following multiplets: These fields are all chosen to be singlets under the U(1) X and SU(3) c gauge groups. To this Higgs sector, we add a Higgs field Ξ with a gauge quantum number (1, 1, b, −b) whose vacuum expectation value (VEV) breaks the U(1) X gauge symmetry. The Higgs sector of the model and the symmetry breaking are governed by the scalar potential [32][33][34][35][36][37][38][39]: are real parameters. Parity symmetry implies that the model has three independent gauge coupling constants: g L = g R = g, g BL and g X . Note that this Higgs potential is invariant under the parity symmetry. The Yukawa couplings of the model are: ij (i, j = 1, 2, 3) are complex Yukawa coupling constants that yield the masses for the fermions of the model, and m ζ is the Dirac mass of the ζ fermion JHEP12(2018)009 which is the dark matter candidate of the model. The model has seesaw mechanisms of Iand II-type for neutrinos as in the usual left-right model.
The gauge symmetry G LR is broken down to SU(3) c × U(1) em by the VEVs of the Higgs fields, which are defined as φ 0 For simplicity, we choose the hierarchy among VEV scales In order to yield the right scale of the electroweak symmetry breaking, we have a relation of GeV and parametrized the VEVs as v 1 = v sin β and v 2 = v cos β with a β-angle. The sequence of the SSB is as follows: first, the SU(2) R × U(1) B−L symmetry is broken by v R to yield the heavy gauge bosons W R and Z R . Next, the U(1) X symmetry is broken by u and a mass of X boson is generated. The electroweak symmetry breaking is completed by v 1 and v 2 . The next section is dedicated to describe masses of the gauge bosons and their interactions with fermions along with phenomenological constraints.

The masses of gauge bosons and the structure of neutral current interactions
After the SSB, the charged gauge bosons acquire their masses as . This mass matrix is diagonalized by a SO(2) transformation: where ϕ is a small mixing angle given by tan(2ϕ) ≃ −2v 2 sin(2β)/v 2 R . Thereby, the mass eigenvalues of W L (which is identified as the SM W boson) and W R bosons are calculated to be We find the charged current interactions among neutrinos-leptons with W L and W R of the form: The neutral gauge boson sector consists of four vector fields: A µ 3 L , A µ 3 R , B µ and C µ . After the SSB, the neutral gauge boson mass matrix can be cast in the form: R B µ C µ , and the mass matrix M 2 is given by Here, we have chosen b = +1/3, for simplicity. In diagonalizing the mass matrix, we carry out an SO(4) transformation V −→Ṽ = R t V , where R is an orthogonal matrix belonging to SO(4). Once the massless photon mode is taken out, the remaining 3 × 3 matrix can be diagonalized by an SO(3) matrix parameterized by three angles. The neutral gauge boson mass eigenvalues in the mass eigenstate basis (Z, Z R , X, A) are given by (upto small mixings that we ignore) In terms of symmetry breaking VEVs, the mass eigenvalues are given by (3.8) The zero mass eigenstate that emerges in eq. (3.7) is identified as the electromagnetic massless photon, which is expressed as a linear combination of the original gauge bosons: Defining e g BL = cos 2θ W sin θ 1 ,

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two parameters, θ 1 and g X , are expressed in terms of g BL as sin θ 1 = 0.43/g BL and g X = g BL tan θ 1 = 0.43g BL / g 2 BL − (0.43) 2 . Thereby, we can use g BL ≥ 0.43 as a free parameter of the model for our analysis in the following sections. In this parameterization, we can express eq. (3.8) as As a benchmark value, we choose the W R mass at the lower limit obtained by the LHC experiment, M W R = 4.4 TeV [11,12], which means v R ≃ 9.6 TeV. This leads to the M Z R value as The mass eigenstates are approximately expressed in terms of the original fields as We ignore small mixings of order ( v v R , u v R , v u ) among them in our discussion below. The neutral bosons, Z, Z R , X and A, interact with a chiral fermion f L(R) (left-or right-handed) of the model as Here, the charge generators, Q Z , Q Z R and Q X , are described as and the electric charge Q em is given by eq. (1.1). All the neutral currents of the model are contained below:

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where J em µ is the usual electromagnetic current, J 0 µ = J 3 L µ − sin 2 θ W J em µ is the Z-neutral current of the GSW model, and the neutral currents of Z R and X are given by More explicitly, the neutral currents can be written in terms of a Dirac fermion f : where f stand for a lepton, a quark or the dark matter particle ζ. The coefficients g f

L(R)
and h f L(R) are defined as Here, we list all the expressions of g f L(R) and h f L(R) in terms of coupling constant g BL and θ W :

21)
and The interaction of neutrinos/leptons with the Z R boson has the contribution of both vector and axial currents,

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and To obtain the decay width formula, it is convenient to write the neutral currents of eq. (3.19) in terms of vector and axial-vector components and we find the partial decay with of the Z R boson into a fermion f as and the vector and axial-vector couplings are defined as g f V /A = 2 g f L ± g f R , respectively. For example, the partial decay widths into charged-leptons and neutrinos (with the rightcomponent contribution) are given by .
where g f = Q f BLg BL withg BL = 1 − (0.43/g BL ) 2 (g BL /2) and Q f BL being the B − L charge of the fermion, and g ζ = 2 ag BL . The effective couplingg BL is a monotonically increasing function of g BL ≥ 0.43 with limits ofg BL → 0 for g BL → 0.43 andg BL ≃ g BL for g 2 BL ≫ (0.43) 2 . Note that even though the electric charge formula implies a lower bound on g BL , the effective X couplingg BL can be smaller. In the following analysis, we useg BL as a free parameter, instead of g BL . Note that the interaction of the X boson with the SM fermions is exactly the same as that of the B − L gauge boson (Z ′ boson) in the minimal B − L model [40][41][42][43][44] when identifyingg BL with the B − L gauge coupling. Since the ζ-charge a is a free parameter, we can use g ζ as a free parameter in our analysis on DM physics.
The ATLAS and CMS collaborations have been searching for a narrow resonance with the dilepton final states (e + e − and µ + µ − ) at the LHC. As a benchmark model, the production of the Z ′ boson of the minimal B − L model has been analyzed by the ATLAS collaboration with a 36.1/fb luminosity and a collider energy of √ s = 13 TeV at the LHC JHEP12(2018)009 Run-2 [29], and the upper bound of the B − L gauge coupling as a function of Z ′ boson mass has been obtained. 3 By identifying the B − L gauge coupling and the Z ′ boson mass withg BL and M X , respectively, we show the current ATLAS bound in figure 1.
In the ATLAS analysis, the Z ′ boson is assumed to decay into only the SM fermions. If the X boson has additional decay modes into new particles, the upper bound must be modified. In our model, the X boson can also decay into a pair of DM particles for 2m ζ < M X . As we can see in the next section, the total decay width of the X boson is very small compared to the X mass, so that the narrow decay width approximation can be justified to evaluate the X production cross section at the LHC Run-2. In the approximation, the cross section of the process qq → X → ℓ + ℓ − at the parton level is described as Hence, if the X boson can decay into a pair of the DM particles, BR(X → ℓ + ℓ − ) becomes smaller and as a result, the upper bound ong BL is increasing. In the presence of the decay of X →ζζ, we scale the result shown in figure 1 by a factor of where Γ SM is the partial decay width of X into all the kinematically allowed SM fermions, and Γ(X →ζζ) is the partial decay width into a pair of the DM particles.  Note that the annihilation cross section in this case is independent ofg BL and this is a crucial difference from Case (i). By takingg BL as small as possible, we can easily avoid the severe constraints from the direct DM detection experiments and the search for X boson at the LHC. As in Case (i), we employ eq. (4.5) and evaluate the DM relic density. In the left panel of figure 4, we show g ζ vs. m ζ along which the observed DM relic abundance is reproduced. In this analysis, we have taken M X = m ζ /3 as an example. We have also calculated the

Conclusions
We have proposed a minimal extension of the Left-Right Symmetric Model (LRSM) for neutrino masses by introducing an extra U(1) X gauge group and a heavy gauge singlet Dirac fermion to provide a unified framework for neutrino masses as well as dark matter. The extra U(1) does contribute to the electric charge formula. The model has an extra neutral gauge boson, X, in addition to the gauge bosons W ± , Z, W R and Z R , which plays a key role in the properties of the dark matter. It also connects the dark sector to the visible SM sector. We discuss the constraints on the mass and coupling of this extra gauge boson (X), by diagonalizing the 4 × 4 neutral gauge boson mass matrix to give the approximate eigenstates. We find that, depending on the relative mass hierarchy between the X-boson and the DM fermion, the allowed parameter space of the DM mass and the DM and SM fermion couplings to X lie in different ranges. The main constraints come from the direct detection bounds as well as the LHC bounds on X production. These constraints can be easily avoided when a pair of dark matter particles dominantly annihilates to a pair of X bosons.
Finally we discuss the prospective bounds on the parameters space from the future experiments. The LUX-ZEPLIN DM experiment [48] for the direct DM search is expected to improve the current upper bound on the nucleon scattering cross section about one order of magnitude: σ SI 3 × 10 −12 pb × m ζ 100 GeV . (6.1) The ATLAS and the CMS collaborations will continue the search for a narrow resonance at the LHC with a luminosity upgrade (High-Luminosity LHC). Since the number of SM background events is very small for a high mass resonance region, we naively scale the current upper bound on the cross section with the 36.1/fb luminosity to a future bound by a factor of 36.1/3000 for a 3000/fb integrated luminosity at the High-Luminosity LHC.
Using the narrow decay width approximation (see eq. (3.29)), we scale the current upper bound ong BL by a factor of 36.1/3000. In figure 5, we show our results with these future prospective bounds.