Dark Gauge U(1) Symmetry for an Alternative Left-Right Model

TL;DR

This paper proposes a gauge-extended alternative left-right model with additional fermions and a singlet scalar, resulting in a two-layer dark matter structure, addressing anomaly cancellation and dark matter identification.

Contribution

It introduces a gauge extension with extra fermions and a singlet scalar to create a consistent anomaly-free model with layered dark matter.

Findings

Model is free of gauge anomalies.

Two layers of dark matter are realized.

Simplifies dark matter identification process.

Abstract

An alternative left-right model of quarks and leptons, where the $SU(2)_R$ lepton doublet $(\nu,l)_R$ is replaced with $(n,l)_R$ so that $n_R$ is not the Dirac mass partner of $\nu_L$, has been known since 1987. Previous versions assumed a global $U(1)_S$ 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 $U(1)_S$. This results in two layers of dark matter, one hidden behind the other.

Tables

Table 1. Table 1 : Particle content of proposed model of dark gauge U ​ ( 1 ) 𝑈 1 U(1) symmetry.

particles	$SU{\left(3\right)}_C$	$SU{\left(2\right)}_L$	$SU{\left(2\right)}_R$	$U{\left(1\right)}_X$	$U{\left(1\right)}_S$
${(u,d)}_L$	3	2	1	1/6	0
${(u,h)}_R$	$3$	1	2	1/6	$-1/2$
$d_R$	$3$	1	1	$-1/3$	0
$h_L$	3	1	1	$-1/3$	$-1$
${(\nu ,l)}_L$	1	2	1	$-1/2$	0
${(n,l)}_R$	1	1	2	$-1/2$	1/2
${\nu }_R$	1	1	1	$0$	0
$n_L$	1	1	1	0	1
$({\varphi }_L^{+},{\varphi }_L^0)$	1	2	1	$1/2$	$0$
$({\varphi }_R^{+},{\varphi }_R^0)$	1	1	2	1/2	1/2
$\eta$	1	2	2	$0$	$-1/2$
$\zeta$	1	1	1	$0$	$1$
${({\psi }_1^0,{\psi }_1^{-})}_R$	1	1	2	$-1/2$	2
${({\psi }_2^{+},{\psi }_2^0)}_R$	1	1	2	$1/2$	1
${\chi }_R^{+}$	1	1	1	1	$-3/2$
${\chi }_R^{-}$	1	1	1	$-1$	$-3/2$
${\chi }_{1R}^0$	1	1	1	0	$-1/2$
${\chi }_{2R}^0$	1	1	1	0	$-5/2$
$\sigma$	1	1	1	0	3

Table 2. Table 2 : Particle content of proposed model under ( T 3 ​ R + S ) × Z 2 subscript 𝑇 3 𝑅 𝑆 subscript 𝑍 2 (T_{3R}+S)\times Z_{2} .

particles	gauge $T_{3R}+S$	global $S^{\prime }$	$Z_3$	$Z_2$
$u,d,\nu ,l$	0	0	1	+
$({\varphi }_L^{+},{\varphi }_L^0),({\eta }_2^{+},{\eta }_2^0),{\varphi }_R^0$	0	0	1	+
$n,{\varphi }_R^{+},\zeta$	1	1	$\omega$	+
$h,({\eta }_1^0,{\eta }_1^{-})$	$-1$	$-1$	${\omega }^2$	+
${\psi }_{2R}^{+},{\chi }_R^{+}$	$3/2,-3/2$	0	$1$	$-$
${\psi }_{1R}^{-},{\chi }_R^{-}$	$3/2,-3/2$	$0$	$1$	$-$
${\psi }_{1R}^0,{\psi }_{2R}^0$	$5/2,1/2$	$1,-1$	$\omega ,{\omega }^2$	$-$
${\chi }_{1R}^0,{\chi }_{2R}^0$	$-1/2,-5/2$	$1,-1$	$\omega ,{\omega }^2$	$-$
$\sigma$	3	0	1	+

Equations

$$\eta=\pmatrix{\eta^{0}_{1}&\eta_{2}^{+}\cr\eta_{1}^{-}&\eta_{2}^{0}},$$

$$0$$

$$({\psi }_{1R}^0{\varphi }_R^{-}+{\psi }_{1R}^{-}{\bar{\varphi }}_R^0){\chi }_R^{+},({\psi }_{2R}^{+}{\varphi }_R^0-{\psi }_{2R}^0{\varphi }_R^{+}){\chi }_R^{-},$$

$$({\psi }_{1R}^0{\varphi }_R^0-{\psi }_{1R}^{-}{\varphi }_R^{+}){\chi }_{2R}^0,({\psi }_{2R}^{+}{\varphi }_R^{-}+{\psi }_{2R}^0{\bar{\varphi }}_R^0){\chi }_{1R}^0.$$

$$⟨{\varphi }_L^0⟩=v_1,⟨{\eta }_2^0⟩=v_2,⟨{\varphi }_R^0⟩=v_R,⟨\sigma ⟩=v_S,$$

$$M_{W_L}^2=\frac{1}{2}{g}_L^2(v_1^2+v_2^2),M_{W_R}^2=\frac{1}{2}{g}_R^2(v_R^2+v_2^2).$$

$$A=\frac{e}{g_L}{W}_{3L}+\frac{e}{g_R}{W}_{3R}+\frac{e}{g_X}X,$$

$$Z$$

$$Z^{\prime }$$

$$M_{ZZ}^2$$

$$M_{Z^{\prime }{Z}^{\prime }}^2$$

$$M_{SS}^2$$

$$M_{Z{Z}^{\prime }}^2$$

$$M_{ZS}^2$$

$$M_{Z^{\prime }S}^2$$

$${\mathcal{L}}_{NC}$$

$$\frac{v_S^2}{v_R^2}=\frac{(g_R^2+g_X^2+g_S^2{)}^2}{36g_S^2(g_R^2+g_X^2-g_S^2)},$$

$$\tan {\theta }_D=\frac{\sqrt{g_R^2+g_X^2}-g_S}{\sqrt{g_R^2+g_X^2}+g_S},$$

$$\pmatrix{D_{1}\cr D_{2}}=\pmatrix{\cos\theta_{D}&\sin\theta_{D}\cr-\sin\theta_{D}&\cos\theta_{D}}\pmatrix{Z^{\prime}\cr S},$$

$$M_{D_1}^2$$

$$M_{D_2}^2$$

$$2g_S=\sqrt{g_R^2+g_X^2},$$

$$\frac{g_X^2}{g_Z^2}=\frac{{\sin }^2{\theta }_W{\cos }^2{\theta }_W}{\cos 2{\theta }_W},\frac{g_S}{g_Z}=\frac{{\cos }^2{\theta }_W}{2\sqrt{\cos 2{\theta }_W}},$$

$$\frac{v_S^2}{v_R^2}=\frac{25}{108},M_{D_2}^2=3M_{D_1}^2=\frac{5{\cos }^4{\theta }_W}{4\cos 2{\theta }_W}{g}_Z^2{v}_R^2.$$

$${\mathcal{L}}_D$$

$$-{\mathcal{L}}_Y$$

$$({\varphi }_{1R}^0{\psi }_{2R}^0-{\psi }_{1R}^{-}{\psi }_{2R}^{+}){\sigma }^{\ast },{\chi }_R^{-}{\chi }_R^{+}\sigma ,{\chi }_{1R}^0{\chi }_{2R}^0\sigma .$$

$${\bar{d}}_R{h}_L\zeta ,{\bar{n}}_L{\nu }_R\zeta$$

$$\eta=\pmatrix{\eta_{1}^{0}&\eta_{2}^{+}\cr\eta_{1}^{-}&\eta_{2}^{0}},~{}~{}~{}\tilde{\eta}=\sigma_{2}\eta^{*}\sigma_{2}=\pmatrix{\bar{\eta}_{2}^{0}&-\eta_{1}^{+}\cr-\eta_{2}^{-}&\bar{\eta}_{1}^{0}},$$

$$V$$

$$2|det(\eta ){|}^2$$

$$({\Phi }_L^{†}{\Phi }_L)Tr({\eta }^{†}\eta )$$

$$({\Phi }_R^{†}{\Phi }_R)Tr({\eta }^{†}\eta )$$

$${\mu }_L^2$$

$${\mu }_{\eta }^2$$

$${\mu }_R^2$$

Full text

Introduction* :

The alternative left-right model [1] of 1987 was inspired by the $E_{6}$ decomposition to the standard $SU(3)_{C}\times SU(2)_{L}\times U(1)_{Y}$ gauge symmetry through an $SU(2)_{R}$ which does not have the conventional assignments of quarks and leptons. Instead of $(u,d)_{R}$ and $(\nu,l)_{R}$ 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 $h_{L}$, $d_{R}$, $n_{L}$, $\nu_{R}$ are singlets.*

This structure allows for the absence of tree-level flavor-changing 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 $S^{\prime}$ 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 $S^{\prime}$, which is now broken to $Z_{3}$. Hence dark matter has two components [5]. They are identified as one Dirac fermion (nontrivial under both $Z_{2}$ and $Z_{3}$) and one complex scalar (nontrivial under $Z_{3}$).

Model* :*

The particle content of our model is given in Table 1, where the scalar $SU(2)_{L}\times SU(2)_{R}$ bidoublet is given by

[TABLE]

with $SU(2)_{L}$ transforming vertically and $SU(2)_{R}$ horizontally. Without $U(1)_{S}$ as a gauge symmetry, the model is free of anomalies without the addition of the $\psi$ and $\chi$ fermions.

In the presence of gauge $U(1)_{S}$, the additional anomaly-free conditions are all satisfied by the addition of the $\psi$ and $\chi$ fermions. The $[SU(3)_{C}]^{2}U(1)_{S}$ anomaly is canceled between $(u,h)_{R}$ and $h_{L}$; the $[SU(2)_{L}]^{2}U(1)_{S}$ anomaly is zero because $(u,d)_{L}$ and $(\nu,l)_{L}$ do not transform under $U(1)_{S}$; the $[SU(2)_{R}]^{2}U(1)_{S}$ and $[SU(2)_{R}]^{2}U(1)_{X}$ anomalies are both canceled by summing over $(u,h)_{R}$, $(n,l)_{R}$, $(\psi_{1}^{0},\psi_{1}^{-})_{R}$, and $(\psi_{2}^{+},\psi_{2}^{0})_{R}$; the addition of $\chi_{R}^{\pm}$ renders the $[U(1)_{X}]^{2}U(1)_{S}$, $U(1)_{X}[U(1)_{S}]^{2}$, $[U(1)_{X}]^{3}$, and $U(1)_{X}$ anomalies zero; and the further addition of $\chi^{0}_{1R}$ and $\chi^{0}_{2R}$ kills both the $[U(1)_{S}]^{3}$ and $U(1)_{S}$ anomalies, i.e.

[TABLE]

Under $T_{3R}+S$, the neutral scalars $\phi_{R}^{0}$ and $\eta_{2}^{0}$ are zero, so that their vacuum expectation values do not break $T_{3R}+S$ which remains as a global symmetry. However, $\langle\sigma\rangle\neq 0$ does break $T_{3R}+S$ and gives masses to $\psi^{0}_{1R}\psi^{0}_{2R}-\psi^{-}_{1R}\psi^{+}_{2R}$, $\chi^{+}_{R}\chi^{-}_{R}$, and $\chi^{0}_{1R}\chi^{0}_{2R}$. These exotic fermions all have half-integral charges [6] under $T_{3R}+S$ and only communicate with the others with integral charges through $W_{R}^{\pm}$, $\sqrt{2}Re(\phi^{0}_{R})$, $\zeta$, and the two extra neutral gauge bosons beyond the $Z$. Some explicit Yukawa terms are

[TABLE]

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. $n,h,W_{R}^{\pm},\phi^{\pm}_{R},\eta_{1}^{\pm},\eta_{1}^{0},\bar{\eta}_{1}^{0}$, $\zeta$, and the underlying exotic fermions with odd $Z_{2}$. Without $\zeta$, a global $S^{\prime}$ symmetry remains. With $\zeta$, because of the $\zeta^{3}\sigma^{*}$ and $\chi^{0}_{1R}\chi^{0}_{1R}\zeta$ terms, the $S^{\prime}$ symmetry breaks to $Z_{3}$.

Let

[TABLE]

then the $SU(3)_{C}\times SU(2)_{L}\times SU(2)_{R}\times U(1)_{X}\times U(1)_{S}$ gauge symmetry is broken to $SU(3)_{C}\times U(1)_{Q}$ with $S^{\prime}$, which becomes $Z_{3}$, as shown in Table 2 with $\omega^{3}=1$. The discrete $Z_{2}$ symmetry is unbroken. Note that the global $S^{\prime}$ 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, $W_{L}^{\pm}$ and $W_{R}^{\pm}$, do not mix because of $S^{\prime}(Z_{3})$, as in the original alternative left-right models. Their masses are given by

[TABLE]

Since $Q=I_{3L}+I_{3R}+X$, the photon is given by

[TABLE]

where $e^{-2}=g_{L}^{-2}+g_{R}^{-2}+g_{X}^{-2}$. Let

[TABLE]

where $g_{Y}^{-2}=g_{R}^{-2}+g_{X}^{-2}$, then the $3\times 3$ mass-squared matrix spanning $(Z,Z^{\prime},S)$ has the entries:

[TABLE]

Their neutral-current interactions are given by

[TABLE]

where $g_{Z}^{2}=g_{L}^{2}+g_{Y}^{2}$ and $\sin^{2}\theta_{W}=g_{Y}^{2}/g_{Z}^{2}$.

In the limit $v^{2}_{1,2}<<v^{2}_{R},v^{2}_{S}$, the mass-squared matrix spanning $(Z^{\prime},S)$ may be simplified if we assume

[TABLE]

and let

[TABLE]

then

[TABLE]

with mass eigenvalues given by

[TABLE]

In addition to the assumption of Eq. (18), let us take for example

[TABLE]

then $\sin\theta_{D}=1/\sqrt{10}$ and $\cos\theta_{D}=3/\sqrt{10}$. Assuming also that $g_{R}=g_{L}$, we obtain

[TABLE]

The resulting gauge interactions of $D_{1,2}$ are given by

[TABLE]

Since $D_{2}$ is $\sqrt{3}$ 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 $\nu_{R}$ 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

[TABLE]

These terms show explicitly that the assignments of Tables 1 and 2 are satisfied.

As for the exotic $\psi$ and $\chi$ fermions, they have masses from the Yukawa terms of Eqs. (4) and (5), as well as

[TABLE]

As a result, two neutral Dirac fermions are formed from the matrix linking $\chi^{0}_{1R}$ and $\psi^{0}_{1R}$ to $\chi^{0}_{2R}$ and $\psi^{0}_{2R}$. Let us call the lighter of these two Dirac fermions $\chi_{0}$, then it is one component of dark matter of our model. The other will be the scalar $\zeta$, to be discussed later. Note that $\chi_{0}$ communicates with $\zeta$ through the allowed $\chi^{0}_{1R}\chi^{0}_{1R}\zeta$ interaction. Note also that the allowed Yukawa terms

[TABLE]

enable the dark fermions $h$ and $n$ to decay into $\zeta$.

Scalar sector* :*

Consider the most general scalar potential consisting of $\Phi_{L,R}$, $\eta$, and $\sigma$. Let

[TABLE]

then

[TABLE]

Note that

[TABLE]

The minimum of $V$ satisfies the conditions

[TABLE]

The $4\times 4$ mass-squared matrix spanning $\sqrt{2}Im(\phi_{L}^{0},\eta_{2}^{0},\phi_{R}^{0},\sigma)$ is then given by

[TABLE]

and that spanning $\sqrt{2}Re(\phi_{L}^{0},\eta_{2}^{0},\phi_{R}^{0},\sigma)$ is

[TABLE]

Hence there are three zero eigenvalues in ${\cal M}^{2}_{I}$ with one nonzero eigenvalue $-\mu_{3}[v_{1}v_{2}/v_{R}+v_{R}(v_{1}^{2}+v_{2}^{2})/v_{1}v_{2}]$ corresponding to the eigenstate $(-v_{1}^{-1},v_{2}^{-1},v_{R}^{-1},0)/\sqrt{v_{1}^{-2}+v_{2}^{-2}+v_{R}^{-2}}$. In ${\cal M}^{2}_{R}$, the linear combination $H=(v_{1},v_{2},0,0)/\sqrt{v_{1}^{2}+v_{2}^{2}}$, is the standard-model Higgs boson, with

[TABLE]

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 direct-search experiments. The addition of the scalar $\zeta$ introduces two important new terms:

[TABLE]

The first term breaks global $S^{\prime}$ to $Z_{3}$, and the second term mixes $\zeta$ with $\eta_{1}^{0}$ through $v_{2}$. We assume the latter to be negligible, so that the physical dark scalar is mostly $\zeta$.

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 $\mu^{-}\mu^{+}$). As noted previously, in our chosen example, $D_{1}$ is the lighter of the two. Hence current search limits for a $Z^{\prime}$ boson are applicable [7, 8]. The $c_{u,d}$ coefficients used in the data analysis are*

[TABLE]

where $B$ is the branching fraction of $Z^{\prime}$ to $e^{-}e^{+}$ and $\mu^{-}\mu^{+}$. 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 $B=1.2\times 10^{-2}$. Based on the 2016 LHC 13 TeV data set, 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. As for the $t$-channel diagrams, they also turn out to be too small. Previous studies where $n$ is chosen as dark matter are now ruled out.

Dark sector* :*

Dark matter is envisioned to have two components. One is a Dirac fermion $\chi_{0}$ which is a mixture of the four neutral fermions of odd $Z_{2}$, and the other is a complex scalar boson which is mostly $\zeta$. The annihilation $\chi_{0}\bar{\chi}_{0}\to\zeta\zeta^{*}$ determines the relic abundance of $\chi_{0}$, and the annihilation $\zeta\zeta^{*}\to HH$, where $H$ is the standard-model Higgs boson, determines that of $\zeta$. The direct $\zeta\zeta^{*}H$ coupling is assumed small to avoid the severe constraint in direct-search experiments.

Let the interaction of $\zeta$ with $\chi_{0}$ be $f_{0}\zeta\chi_{0R}\chi_{0R}+H.c.$, then the annihilation cross section of $\chi_{0}\bar{\chi}_{0}$ to $\zeta\zeta^{*}$ times relative velicity is given by

[TABLE]

Let the effective interaction strength of $\zeta\zeta^{*}$ with $HH$ be $\lambda_{0}$, then the annihilation cross section of $\zeta\zeta^{*}$ to $HH$ times relative velicity is given by

[TABLE]

Note that $\lambda_{0}$ is the sum over several interactions. The quartic coupling $\lambda_{\zeta H}$ is assumed negligible, to suppress the trilinear $\zeta\zeta^{*}H$ coupling which contributes to the elastic $\zeta$ scattering cross section off nuclei. However, the trilinear couplings $\zeta\zeta^{*}Re(\phi_{R}^{0})$ and $Re(\phi_{R}^{0})HH$ are proportional to $v_{R}$, and the trilinear couplings $\zeta\zeta^{*}Re(\sigma)$ and $Re(\sigma)HH$ are proportional to $v_{S}$. Hence their effective contributions to $\lambda_{0}$ are proportional to $v_{R}^{2}/m^{2}[\sqrt{2}Re(\phi_{R}^{0})]$ and $v_{S}^{2}/m^{2}[\sqrt{2}Re(\sigma)]$, which are not suppressed.

As a rough estimate, we will assume that

[TABLE]

to satisfy the condition of dark-matter relic abundance [9] of the Universe. For given values of $m_{\zeta}$ and $m_{\chi_{0}}$, the parameters $\lambda_{0}$ and $f_{0}$ are thus constrained. We show in Fig. 1 the plots of $\lambda_{0}$ versus $f_{0}$ for $m_{\zeta}=150$ GeV and various values of $m_{\chi_{0}}$. Since $m_{\zeta}$ is fixed at 150 GeV, $\lambda_{0}$ is also fixed for a given fraction of $\Omega_{\zeta}/\Omega_{DM}$. To adjust for the rest of dark matter, $f_{0}$ must then vary as a function of $m_{\chi_{0}}$ according to Eq. (44).

As for direct detection, both $\chi_{0}$ and $\zeta$ have possible interactions with quarks through the gauge bosons $D_{1,2}$ and the standard-model Higgs boson $H$. They are suppressed by making the $D_{1,2}$ masses heavy, and the $H$ couplings to $\chi_{0}$ and $\zeta$ small. In our example with $m_{\zeta}=150$ GeV, let us choose $m_{\chi_{0}}=500$ Gev and the relic abundances of both to be equal. From Fig. 1, these choices translate to $\lambda_{0}=0.12$ and $f_{0}=0.56$.

Consider first the $D_{1,2}$ interactions. Using Eq. (26), we obtain

[TABLE]

The effective $\zeta$ elastic scattering cross section through $D_{1,2}$ is then completely determined as a function of the $D_{1}$ mass (because $M_{D_{2}}=\sqrt{3}M_{D_{1}}$ in our example), i.e.

[TABLE]

Using the latest LUX result [10] and Eq. (25), we obtain $v_{R}>35$ TeV which translates to $M_{D_{1}}>18$ TeV, and $M_{W_{R}}>16$ TeV.

The $\bar{\chi_{0}}\gamma_{\mu}\chi_{0}$ couplings to $D_{1,2}$ depend on the $2\times 2$ mass matrix linking $(\chi_{1},\psi_{1})$ to $(\chi_{2},\psi_{2})$ which has two mixing angles and two mass eigenvalues, the lighter one being $m_{\chi_{0}}$. By adjusting these parameters, it is possible to make the effective $\chi_{0}$ interaction with xenon negligibly small. Hence there is no useful limit on the $D_{1}$ mass in this case.

Direct search also constrains the coupling of the Higgs boson to $\zeta$ (through a possible trilinear $\lambda_{\zeta H}\sqrt{2}v_{H}\zeta^{*}\zeta$ interaction) or $\chi_{0}$ (through an effective Yukawa coupling $\epsilon$ from $H$ mixing with $\sigma_{R}$ and $\phi_{R}^{0}$). Let their effective interactions with quarks through $H$ exchange be given by

[TABLE]

where $f_{q}=m_{q}/\sqrt{2}v_{H}=m_{q}/(246~{}{\rm GeV})$. The spin-independent direct-detection cross section per nucleon in the former is given by

[TABLE]

where $\mu_{\zeta}=m_{\zeta}M_{A}/(m_{\zeta}+M_{A})$ is the reduced mass of the dark matter, and [11]

[TABLE]

with [12]

[TABLE]

For $m_{\zeta}=150$ GeV, we have

[TABLE]

Using $A=131$, $Z=54$, and $M_{A}=130.9$ atomic mass units for the LUX experiment [10], and twice the most recent bound of $2\times 10^{-46}~{}cm^{2}$ (because $\zeta$ is assumed to account for only half of the dark matter) at this mass, we find

[TABLE]

As noted earlier, this is negligible for considering the annihilation cross section of $\zeta$ to $H$.

For the $H$ contribution to the $\chi_{0}$ elastic cross section off nuclei, we replace $m_{\zeta}$ with $m_{\chi_{0}}=500$ GeV in Eq. (51) and $\lambda_{\zeta H}/2m_{\zeta}$ with $\epsilon/\sqrt{2}v_{H}$ in Eq. (52). Using the experimental data at 500 GeV, we obtain the bound.

[TABLE]

From the above discussion, it is clear that our model allows for the discovery of dark matter in direct-search experiments in the future if these bounds are only a little above the actual values of $\lambda_{\zeta H}$ and $\epsilon$.

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 $\chi_{0}\sim(\omega,-)$ and the other the complex scalar $\zeta\sim(\omega,+)$ under $Z_{3}\times Z_{2}$. 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 $\zeta$ is a significant fraction of dark matter. If $\chi_{0}$ 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 $\zeta$ through the direct $\bar{d}_{R}h_{L}\zeta$ 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 [13] based on the $\sqrt{s}=13$ TeV data at the LHC with an integrated luminosity of 35.9 fb-1 for a single scalar quark.

If the $\bar{d}_{R}h_{L}\zeta$ coupling is very small, then $h$ may also decay significantly to $u$ and a virtual $W_{R}^{-}$, with $W_{R}^{-}$ becoming $\bar{n}l^{-}$, and $\bar{n}$ becoming $\bar{\nu}\zeta^{*}$. 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.

Acknowledgement* :  This work was supported in part by the U. S. Department of Energy Grant No. DE-SC0008541.*