Dark Gauge U(1) symmetry for an alternative left–right model

An alternative left–right model of quarks and leptons, where the SU(2)R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$SU(2)_R$$\end{document} lepton doublet (ν,l)R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\nu ,l)_R$$\end{document} is replaced with (n,l)R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n,l)_R$$\end{document} so that nR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_R$$\end{document} is not the Dirac mass partner of νL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu _L$$\end{document}, has been known since 1987. Previous versions assumed a global U(1)S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U(1)_S$$\end{document} symmetry to allow n to be identified as a dark-matter fermion. We propose here a gauge extension by the addition of extra fermions to render the model free of gauge anomalies, and just one singlet scalar to break U(1)S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U(1)_S$$\end{document}. This results in two layers of dark matter, one hidden behind the other.


Introduction
The alternative left-right model [1] of 1987 was inspired by the E 6 decomposition to the standard SU (3) C × SU (2) L × U (1) Y gauge symmetry through an SU (2) R , which does not have the conventional assignments of quarks and leptons. Instead of (u, d) R and (ν, l) R as doublets under SU (2) R , a new quark h and a new lepton n per family are added so that (u, h) R and (n, e) R are the SU (2) R doublets, and h L , d R , n L , ν R are singlets.
This structure allows for the absence of tree-level flavorchanging neutral currents (unavoidable in the conventional model), as well as the existence of dark matter. The key new ingredient is a U (1) S symmetry, which breaks together with SU (2) R , such that a residual global S symmetry remains for the stabilization of dark matter. Previously [2][3][4], this U (1) S was assumed to be global. We show in this paper how it may be promoted to a gauge symmetry. To accomplish this, new fermions are added to render the model free of gauge anomalies. The resulting theory has an automatic discrete Z 2 symmetry which is unbroken as well as the global S , which is now broken to Z 3 . Hence dark matter has two components [5]. They are identified as one Dirac fermion (nontrivial under a e-mail: ma@phyun8.ucr.edu both Z 2 and Z 3 ) and one complex scalar (nontrivial under Z 3 ).
In Sect. 2 we make a digression to the historical perspective which motivated this study. In Sect. 3 our model is described, with a complete list of its particle content. In Sect. 4 the gauge sector is shown in detail. In Sect. 5 the fermions are discussed with details of how they obtain masses. In Sect. 6 we deal with the scalars and show how the desirable pattern of symetry breaking is obtained. In Sect. 7 we discuss the present phenomenological constraints on the new Z bosons and would-be dark-matter candidates. In Sect. 8 we show an example of two viable dark-matter candidates, both in terms of relic abundance and direct detection. In Sect. 9 we conclude.

Motivation and historical perspective
This section is for those who are unfamiliar with, but interested in the historical perspective which motivated our study. In the beginning, the idea of an SU (2) L × SU (2) R electroweak extension of the standard model (SM), which is based only on SU (2) L , was very attractive, because it restores left-right symmetry to the interactions of the quarks and leptons. In the conventional approach, (u, d) i L are SU (2) L doublets and (u, d) j R are SU (2) R doublets. To allow them to have masses, a scalar bidoublet is needed, so thatū i L u j R couple to δ 0 1 andd i L d j R couple to δ 0 2 , thereby obtaining masses from the vacuum expectation values of the two neutral scalars. However, because of the peculiarity of SU (2) doublets, the bidoublet transforms identically as . Hence δ 2 contributes to the u mass matrix, and δ 1 contributes to the d mass matrix. In other words, each quark sector gets its masses from two different Higgs particles. This means that flavor changing neutral currents (FCNC) are unavoidable at tree level through neutral Higgs exchange. This is a very strong constraint on the masses of these particles, of order 10-100 TeV. As such they are not likely to be observable at the Large Hadron Collider (LHC). On the general issue of FCNC, they are, of course, present in the SM, but only at the loop level, and they are known to be small and consistent with experimental data. In any extension of the SM, they may occur at tree level, and if so the scalar particles in question are required to be very heavy and out of reach of the LHC. It is thus a valid question to ask whether a model beyond the SM may be constructed with the absence of tree-level FCNC, so that it may have new scalars which are light enough to be discovered in addition to the SM Higgs boson of 125 GeV.
To distinguish˜ from , an extra symmetry is needed. This is what happens in supersymmetry, but then the u quark mass matrix must be proportional to the d quark mass matrix, which disagrees with data. The solution to this conundrum was pointed out 30 years ago [1]. It was discovered in the context of superstring-inspired E 6 models, but applicable to the SU (2) L × SU (2) R case [2,3]. The idea is to add another quark h to each family which has the same charge as d, i.e. −1/3. Both h L and h R are singlets in the SM, but they are distinguished from d L and d R in their SU (2) R assignments, i.e.
To forbid the termh L d R , a global U (1) S symmetry is added which also distinguishes from˜ . In this way, the d mass comes from an SU (2) L Higgs doublet, the h mass comes from an SU (2) R Higgs doublet, and the u mass comes from only δ 0 1 whereas δ 0 2 has no vacuum expectation value. Thus the model is guaranteed the absence of tree-level FCNC. It was realized a few years ago [2,3] that this extra U (1) S also serves the purpose of a dark symmetry, because even though it is broken, the combination T 3R + S or T 3R − S may remain unbroken and protects the condition δ 0 2 = 0. In other words, the symmetry which allows us to solve the FCNC conundrum has now been connected to that of dark matter. Contrast this with most models of dark matter, where the existence of the dark symmetry is completely ad hoc, and unrelated to any other symmetry of the original model. This we believe is a good motivation for studying alternative left-right models. The logical next step is to ask the question whether it is possible for this U (1) S to be gauged. What follows is a simple example of how it can be done and the resulting consequences.
The particle content of our model is given in Table 1.
Without U (1) S as a gauge symmetry, the model is free of anomalies without the addition of the ψ and χ fermions. In the presence of gauge U (1) S , the additional anomalyfree conditions are all satisfied by the addition of the ψ and χ fermions. 3 , and U (1) X anomalies zero; and the further addition of χ 0 1R and χ 0 2R kills both the [U (1) S ] 3 and the U (1) S anomalies, i.e.
The scalar SU (2) L × SU (2) R bidoublet is given by with SU (2) L transforming vertically and SU (2) R horizontally. Under T 3R +S, the neutral scalars φ 0 R and η 0 2 are zero, so that their vacuum expectation values do not break T 3R + S, which remains as a global symmetry. However, σ = 0 does break T 3R + S and gives masses to ψ 0 These exotic fermions all have halfintegral charges [6] under T 3R + S and only communicate with the others with integral charges through W ± R , √ 2Re(φ 0 R ), ζ , and the two extra neutral gauge bosons beyond the Z . Some explicit Yukawa terms are This dichotomy of particle content results in an additional unbroken symmetry of the Lagrangian, i.e. discrete Z 2 under which the exotic fermions are odd. Hence dark matter has two layers: those with nonzero T 3R + S and even Z 2 , i.e. n, h, W ± R , φ ± R , η ± 1 , η 0 1 ,η 0 1 , ζ , and the underlying exotic fermions with odd Z 2 . Without ζ , a global S symmetry remains. With ζ , because of the ζ 3 σ * and χ 0 1R χ 0 1R ζ terms, the S symmetry breaks to Z 3 .
Let Table 2 with ω 3 = 1. The discrete Z 2 symmetry is unbroken. Note that the global S assignments for the exotic fermions are not T 3R + S because of v S , which breaks the gauge U (1) S by 3 units.

Gauge sector
Consider now the masses of the gauge bosons. The charged ones, W ± L and W ± R , do not mix because of S (Z 3 ), as in the original alternative left-right models. Their masses are given by Since Q = I 3L + I 3R + X , the photon is given by where where g −2 Y = g −2 R + g −2 X , then the 3 × 3 mass-squared matrix spanning (Z , Z , S) has the entries: Their neutral-current interactions are given by and let then with mass eigenvalues given by In addition to the assumption of Eq. (18), let us take for example then sin θ D = 1/ √ 10 and cos θ D = 3/ √ 10. Assuming also that g R = g L , we obtain The resulting gauge interactions of D 1,2 are given by Since D 2 is √ 3 times heavier than D 1 in this example, the latter would be produced first in pp collisions at the Large Hadron Collider (LHC).

Fermion sector
All fermions obtain masses through the four vacuum expectation values of Eq. (6) except ν R , which is allowed to have an invariant Majorana mass. This means that neutrino masses may be small from the usual canonical seesaw mechanism. The various Yukawa terms for the quark and lepton masses are These terms show explicitly that the assignments of Tables 1 and 2 are satisfied. As for the exotic ψ and χ fermions, they have masses from the Yukawa terms of Eqs. (4) and (5), as well as from As a result, two neutral Dirac fermions are formed from the matrix linking χ 0 1R and ψ 0 1R to χ 0 2R and ψ 0 2R . Let us call the lighter of these two Dirac fermions χ 0 , then it is one component of dark matter of our model. The other will be the scalar ζ , to be discussed later. Note that χ 0 communicates with ζ through the allowed χ 0 1R χ 0 1R ζ interaction. Note also that the allowed Yukawa terms enable the dark fermions h and n to decay into ζ .

Scalar sector
Consider the most general scalar potential consisting of L ,R , η, and σ . Let Note that The minimum of V satisfies the conditions The 4 × 4 mass-squared matrix spanning √ 2I m(φ 0 L , η 0 2 , φ 0 R , σ ) is then given by Hence there are three zero eigenvalues in is the standard-model Higgs boson, with The other three scalar bosons are much heavier, with suppressed mixing to H , which may all be assumed to be small enough to avoid the constraints from dark-matter directsearch experiments. The addition of the scalar ζ introduces two important new terms: The first term breaks global S to Z 3 , and the second term mixes ζ with η 0 1 through v 2 . We assume the latter to be negligible, so that the physical dark scalar is mostly ζ .

Present phenomenological constraints
Many of the new particles of this model interact with those of the standard model. The most important ones are the neutral D 1,2 gauge bosons, which may be produced at the LHC through their couplings to u and d quarks, and decay to charged leptons (e − e + and μ − μ + ). As noted previously, in our chosen example, D 1 is the lighter of the two. Hence current search limits for a Z boson are applicable [7,8]. The c u,d coefficients used in the data analysis are where B is the branching fraction of Z to e − e + and μ − μ + . Assuming that D 1 decays to all the particles listed in Table 2, except for the scalars which become the longitudinal components of the various gauge bosons, we find B = 1.2 × 10 −2 . Based on the 2016 LHC 13 TeV data set from ATLAS [9], this translates to a bound of about 4 TeV on the D 1 mass. The would-be dark-matter candidate n is a Dirac fermion which couples to D 1,2 , which also couples to quarks. Hence severe limits exist on the masses of D 1,2 from underground direct-search experiments as well. The annihilation cross section of n through D 1,2 would then be too small, so that its relic abundance would be too big for it to be a dark-matter candidate. Its annihilation at rest through s-channel scalar exchange is p-wave suppressed and does not help, barring of course any accidental resonance enhancement. As for the t-channel diagrams, they also turn out to be too small. Suggestions of previous studies [2,3] where n is chosen as dark matter are now ruled out.
determines that of ζ . The direct ζ ζ * H coupling is assumed small to avoid the severe constraint in direct-search experiments.
Let the interaction of ζ with χ 0 be f 0 ζ χ 0R χ 0R + H.c., then the annihilation cross section of χ 0χ0 to ζ ζ * times relative velocity is given by This determines the relic abundance of χ 0 . As the Universe cools below m χ 0 , χ 0 decouples from the thermal bath. We assume that m ζ is much below m χ 0 so that χ 0 is essentialy frozen out at m ζ . The relic abundance of ζ is then mostly determined by ζ ζ * → H H. Let the effective interaction strength of ζ ζ * with H H be λ 0 , then the annihilation cross section of ζ ζ * to H H times relative velocity is given by Note that λ 0 is the sum over several interactions. The quartic coupling λ ζ H is assumed negligible, to suppress the trilinear ζ ζ * H coupling which contributes to the elastic ζ scattering cross section off nuclei. However, the trilinear couplings ζ ζ * Re(φ 0 R ) and Re(φ As a rough estimate, we will assume that to satisfy the condition of dark-matter relic abundance [10] of the Universe. For given values of m ζ and m χ 0 , the parameters λ 0 and f 0 are thus constrained. We show in Fig. 1   Consider first the D 1,2 interactions. Using Eq. (26), we obtain The effective ζ elastic scattering cross section through D 1,2 is then completely determined as a function of the D 1 mass (because M D 2 = √ 3M D 1 in our example), i.e.
Using the most recent XENON result [11] at m ζ = 150 GeV for which σ < 2 × 10 −46 cm 2 and Eq. (25), we obtain v R > 35 TeV which translates to M D 1 > 18 TeV, and M W R > 16 TeV. These are a few percent more restrictive than the most recent LUX result [12]. Theχ 0 γ μ χ 0 couplings to D 1,2 depend on the 2 × 2 mass matrix linking (χ 1 , ψ 1 ) to (χ 2 , ψ 2 ), which has two mixing angles and two mass eigenvalues, the smaller one being m χ 0 . By adjusting these parameters, it is possible to make the effective χ 0 interaction to any particular nucleus through D 1,2 negligibly small. Hence there is no useful limit on the D 1 mass in this case. Note that the amplitude cancellation here is through D 1,2 and not necessarily through u and d quarks (which are not adjustable in this model), as would be necessary in models with only one vector mediator.
Direct search also constrains the coupling of the Higgs boson to ζ (through a possible trilinear λ ζ H √ 2v H ζ * ζ interaction) or χ 0 (through an effective Yukawa coupling from H mixing with σ R and φ 0 R ). Let their effective interactions with quarks through H exchange be given by where f q = m q / √ 2v H = m q /(246 GeV). The spinindependent direct-detection cross section per nucleon in the former is given by where μ ζ = m ζ M A /(m ζ + M A ) is the reduced mass of the dark matter, and [13] with [14] For m ζ = 150 GeV, we have Using A = 131, Z = 54, and M A = 130.9 atomic mass units for the XENON experiment [11], and twice the most recent bound of 2 × 10 −46 cm 2 (at m ζ = 150 GeV) because ζ is assumed to account for only half of the dark matter) at this mass, we find As noted earlier, this is negligible for considering the annihilation cross section of ζ to H . For the H contribution to the χ 0 elastic cross section off nuclei, we replace m ζ with m χ 0 = 500 GeV in Eq. (51) and λ ζ H /2m ζ with / √ 2v H in Eq. (52). Using the experimental data at 500 GeV, we obtain the bound.
From the above discussion, it is clear that it is possible for future improvements in direct-search experiments to yield positive results within the framework of our model.

Conclusion and outlook
In the context of the alternative left-right model, a new gauge U (1) S symmetry has been proposed to stabilize dark matter. This is accomplished by the addition of a few new fermions to cancel all the gauge anomalies, as shown in Table 1. As a result of this particle content, an automatic unbroken Z 2 symmetry exists on top of U (1) S , which is broken to a conserved residual Z 3 symmetry. Thus dark matter has two components. One is the Dirac fermion χ 0 ∼ (ω, −) and the other the complex scalar ζ ∼ (ω, +) under Z 3 × Z 2 . We have shown how they may account for the relic abundance of dark matter in the Universe, and satisfy present experimental search bounds. Whereas we have no specific prediction for discovery in direct-search experiments, our model will be able to accommodate any positive result in the future, just like many other existing proposals. To single out our model, many additional details must also be confirmed. Foremost are the new gauge bosons D 1,2 . Whereas the LHC bound is about 4 TeV, the direct-search bound is much higher, provided that ζ is a significant fraction of dark matter. If χ 0 dominates instead, the adjustment of free parameters of our model can lower this bound to below 4 TeV. In that case, future D 1,2 observations are still possible at the LHC as more data become available.
Another is the exotic h quark which is easily produced if kinematically allowed. It would decay to d and ζ through the directd R h L ζ coupling of Eq. (29). Assuming that this branching fraction is 100%, the search at the LHC for 2 jets plus missing energy puts a limit on m h of about 1.0 TeV, as reported by the CMS Collaboration [15] based on the √ s = 13 TeV data at the LHC with an integrated luminosity of 35.9 fb −1 for a single scalar quark.
If thed R h L ζ coupling is very small, then h may also decay significantly to u and a virtual W − R , with W − R becomingnl − , andn becomingνζ * . This has no analog in the usual searches for supersymmetry or the fourth family because W R is heavy (> 16 TeV). To be specific, the final states of 2 jets plus l − 1 l + 2 plus missing energy should be searched for. As more data are accumulated at the LHC, such events may become observable.