Kinetic mixing effect in noncommutative $B-L$ gauge theory

TL;DR

This paper investigates the kinetic mixing effects in a noncommutative $B-L$ gauge theory, exploring how these mixings influence electroweak precision tests and the model's physical predictions.

Contribution

It constructs a detailed noncommutative $B-L$ gauge model and analyzes the interplay between kinetic mixing and symmetry breaking effects.

Findings

Kinetic mixing impacts electroweak precision observables.

The model provides insights into neutrino mass and dark matter.

Detailed physical implications of kinetic mixing are derived.

Abstract

It is well established that the $SU(P)_L$ gauge symmetry for $P\geq 3$ can address the question of fermion generation number due to the anomaly cancellation, but it neither commutes nor closes algebraically with electric and baryon-minus-lepton charges. Hence, two $U(1)$ factors that determine such charges are required, yielding a complete gauge symmetry, $SU(P)_L\otimes U(1)_X\otimes U(1)_N$, apart from the color group. The resulting theory manifestly provides neutrino mass, dark matter, inflation, and baryon asymmetry of the universe. Furthermore, this gauge structure may present kinetic mixing effects associated to the $U(1)$ gauge fields, which affect the electroweak precision test such as the $\rho$ parameter and $Z$ couplings as well as the new physics processes. We will construct the model, examine the interplay between the kinetic mixing and those due to the symmetry breaking, and obtain the physical results in detail.

Figures

Equations

$$SU(3{)}_C\otimes SU(4{)}_L\otimes U(1{)}_X\otimes U(1{)}_N.$$

$$Q=T_3+\beta T_8+\gamma T_{15}+X,$$

$$B-L=b{T}_8+c{T}_{15}+N,$$

$$[Q,T_1\pm i{T}_2]=\pm (T_1\pm i{T}_2),$$

$$[Q,T_4\pm i{T}_5]=\mp q(T_4\pm i{T}_5),$$

$$[Q,T_6\pm i{T}_7]=\mp (1+q)(T_6\pm i{T}_7),$$

$$[Q,T_9\pm i{T}_{10}]=\mp p(T_9\pm i{T}_{10}),$$

$$[Q,T_{11}\pm i{T}_{12}]=\mp (1+p)(T_{11}\pm i{T}_{12}),$$

$$[Q,T_{13}\pm i{T}_{14}]=\mp (p-q)(T_{13}\pm i{T}_{14}),$$

$$[B-L,T_4\pm i{T}_5]=\mp (1+n)(T_4\pm i{T}_5),$$

$$[B-L,T_6\pm i{T}_7]=\mp (1+n)(T_6\pm i{T}_7),$$

$$[B-L,T_9\pm i{T}_{10}]=\mp (1+m)(T_9\pm i{T}_{10}),$$

$$[B-L,T_{11}\pm i{T}_{12}]=\mp (1+m)(T_{11}\pm i{T}_{12}),$$

$$[B-L,T_{13}\pm i{T}_{14}]=\mp (m-n)(T_{13}\pm i{T}_{14}),$$

$$\displaystyle\psi_{aL}\equiv\left(\begin{array}[]{c}\nu\\ e\\ E^{q,n}\\ F^{p,m}\end{array}\right)_{aL}\sim\left(1,4,\frac{p+q-1}{4},\frac{m+n-2}{4}\right),$$

$$\displaystyle Q_{\alpha L}\equiv\left(\begin{array}[]{c}d\\ -u\\ J^{-q-1/3,-n-2/3}\\ K^{-p-1/3,-m-2/3}\end{array}\right)_{\alpha L}\sim\left(3,4^{*},-\frac{p+q+1/3}{4},-\frac{m+n+2/3}{4}\right),$$

$$\displaystyle Q_{3L}\equiv\left(\begin{array}[]{c}u\\ d\\ J^{q+2/3,n+4/3}\\ K^{p+2/3,m+4/3}\end{array}\right)_{3L}\sim\left(3,4,\frac{p+q+5/3}{4},\frac{m+n+10/3}{4}\right),$$

$${\nu }_{aR}\sim \left(1,1,0,-1\right),e_{aR}\sim \left(1,1,-1,-1\right),$$

$$E_{aR}\sim \left(1,1,q,n\right),F_{aR}\sim \left(1,1,p,m\right),$$

$$u_{aR}\sim \left(3,1,2/3,1/3\right),d_{aR}\sim \left(3,1,-1/3,1/3\right),$$

$$J_{\alpha R}\sim \left(3,1,-q-1/3,-n-2/3\right),K_{\alpha R}\sim \left(3,1,-p-1/3,-m-2/3\right),$$

$$J_{3R}\sim \left(3,1,q+2/3,n+4/3\right),K_{3R}\sim \left(3,1,p+2/3,m+4/3\right),$$

$$\eta$$

$$\rho$$

$$\chi$$

$$\Xi$$

$$\varphi$$

$$\mathcal{L}={\mathcal{L}}_{\mathrm{kinetic}}+{\mathcal{L}}_{\mathrm{Yukawa}}-V,$$

$${\mathcal{L}}_{\mathrm{kinetic}}$$

$$D_{\mu }={\partial }_{\mu }+i{g}_s{T}_r{G}_{r\mu }+ig{T}_i{A}_{i\mu }+i{g}_{X}X{B}_{\mu }+i{g}_{N}N{C}_{\mu },$$

$${\mathcal{L}}_{\mathrm{Yukawa}}$$

$$V$$

$$⟨\eta ⟩$$

$$P=(-1{)}^{3(B-L)+2s}=(-1{)}^{3(b{T}_8+c{T}_{15}+N)+2s}$$

$$[m_E{]}_{ab}=-h_{ab}^E\frac{w}{\sqrt{2}},[m_F{]}_{ab}=-h_{ab}^F\frac{V}{\sqrt{2}},$$

$$[m_J{]}_{33}=-h_{33}^J\frac{w}{\sqrt{2}},[m_J{]}_{\alpha \beta }=-h_{\alpha \beta }^J\frac{w}{\sqrt{2}},$$

$$[m_K{]}_{33}=-h_{33}^K\frac{V}{\sqrt{2}},[m_K{]}_{\alpha \beta }=-h_{\alpha \beta }^K\frac{V}{\sqrt{2}},$$

$$[m_e{]}_{ab}=-h_{ab}^e\frac{v}{\sqrt{2}},$$

$$[m_u{]}_{3a}=-h_{3a}^u\frac{u}{\sqrt{2}},[m_u{]}_{\alpha a}=h_{\alpha a}^u\frac{v}{\sqrt{2}},$$

$$[m_d{]}_{3a}=-h_{3a}^d\frac{v}{\sqrt{2}},[m_d{]}_{\alpha a}=-h_{\alpha a}^d\frac{u}{\sqrt{2}},$$

$$m_{\nu }^L\simeq -m_{\nu }(m_{\nu }^R{)}^{-1}(m_{\nu }{)}^T\sim u^2/\Lambda ,$$

$${\mathcal{L}}_{\mathrm{kinetic}}\supset -\frac{1}{4}{B}_{\mu \nu }^2-\frac{1}{4}{C}_{\mu \nu }^2-\frac{\delta }{2}{B}_{\mu \nu }{C}^{\mu \nu }=-\frac{1}{4}(B_{\mu \nu }+\delta C_{\mu \nu }{)}^2-\frac{1}{4}(1-{\delta }^2)C_{\mu \nu }^2,$$

$$B^{\prime }=B+\delta C,C^{\prime }=\sqrt{1-{\delta }^2}C.$$

$$D_{\mu }\supset i{g}_{X}X{B}_{\mu }+i{g}_{N}N{C}_{\mu }=i{g}_{X}X{B}_{\mu }^{\prime }+\frac{i}{\sqrt{1-{\delta }^2}}(g_{N}N-g_{X}X\delta )C_{\mu }^{\prime },$$

Full text

I Introduction

The standard model of fundamental particles and interactions has been very successful in describing the observed phenomena, but it is incomplete. First of all, the experimental evidences of neutrino oscillations caused by nonzero small neutrino masses and flavor mixing require new physics beyond the standard model neutrino . Additionally, the cosmological challenges of particle physics such as inflation, dark matter, and baryon asymmetry also acquire the standard model extension pdg2018 . Hence, it is worthwhile to look for a theory that addresses all these puzzles.

The standard model actually contains a hidden/accident symmetry $U(1)_{B-L}$. If one includes, e.g., three right-handed neutrinos it behaves as a gauge symmetry free from all the anomalies. The resulting theory based on $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)_{B-L}$ can provide consistent neutrino masses via induced seesaw mechanism seesaw . This theory also generates suitable baryon asymmetry converted from the leptogenesis resulting from the seesaw mechanism leptog . However, within this framework, it is not naturally to understand dark matter. Indeed, a matter parity can be induced as residual gauge symmetry due to the $U(1)_{B-L}$ breaking. However, the theory does not contain any odd field responsible for dark matter candidate. Let us note that the Majoron associated with $B-L$ breaking is actually eaten by the new neutral gauge boson, which should rapidly decay into quarks and leptons. The $B-L$ Higgs field and right-handed neutrinos are also unstable, since they decay to ordinary particles.

In this work, we discuss a class of models based upon gauge symmetry, $SU(3)_{C}\otimes SU(P)_{L}\otimes U(1)_{X}\otimes U(1)_{N}$, called 3-$P$-1-1, for $P=3,4$. Here, $SU(2)_{L}$ is extended to $SU(P)_{L}$ which offers a natural solution for the question of generation number 331 . It is easily verified that the electric charge $Q$ and baryon-minus-lepton charge $B-L$ neither commute nor close algebraically with $SU(P)_{L}$ 3311 ; dong2015 . Hence, the two Abelian factors $U(1)_{X,N}$ are resulted from algebraic closure condition, in which the new charges $X$ and $N$ are related to $Q$ and $B-L$ via the Cartan generators of $SU(P)_{L}$, respectively. Besides the answer of generation number, the model manifestly accommodates dark matter which is unified with normal matter to form $SU(P)_{L}$ multiplets. This is a consequence of the noncommutative $B-L$ symmetry and matter partiy as a residual gauge symmetry. Such dark fields have “wrong” $B-L$ charge in comparison to the standard model definition, that is old under the matter parity, providing dark matter candidates. They may be a fermion, scalar or gauge boson. The abundance of dark matter observed today can either be thermally produced as a WIMP or results from a standard leptogenesis similarly to the baryon asymmetry a3311 . Therefore, in the second case both the dark and normal matter asymmetries are produced due to the CP-violating decay of the lightest right-handed neutrino. In a scenario, the $U(1)_{N}$ breaking field successfully inflates the early universe, and its decay reheats the universe producing such right-handed neutrinos, as desirable a3311 .

The 3-3-1-1 model has been extensively investigated in the literature 3311 ; dong2015 ; a3311 , but the 3-4-1-1 model has not considered yet. In this work, we construct the 3-4-1-1 model with general fermion and scalar contents, obtain the matter parity, and interpret dark matter candidates. Since the theory contains two $U(1)$ factors, the kinetic mixing between the corresponding gauge bosons is not avoidable kineticmixing . Therefore, we diagonalize the gauge boson sector when including the kinetic mixing term. The effect of the kinetic mixing is present in the $\rho$ parameter and the coupling of $Z$ with fermions, which can alter the electroweak precision test. It significantly modifies the neutral meson mixings and rare meson decays. The last aim of this work is to probe the new physics of the model at the LHC. This work also revisits the kinetic mixing effect in the 3-3-1-1 model, which was previously studied dong2016 .

The rest of this work is organized as follows. In Sec. II, we introduce the model and show dark matter. In Sec. III, we diagonalize the gauge sector. In Sec. IV, we examine the $\rho$ parameter, mixing parameters, and the $Z$ couplings. In Sec. V, we investigate the FCNCs. The search for the new physics is presented in Sec. VI. In Sec. VII, the kinetic mixing effect in a previous study is revisited. Finally, we conclude this work in Sec. VIII.

II The model

In this section we propose the 3-4-1-1 model, while the 3-3-1-1 model dong2015 ; dong2016 was well established and skipped.

II.1 Gauge symmetry

As stated, the gauge symmetry is given by

[TABLE]

The electric and baryon-minus-lepton charges are embedded as

[TABLE]

where $T_{i}$ ($i=1,2,3,...,15$), $X$, and $N$ are $SU(4)_{L}$, $U(1)_{X}$, and $U(1)_{N}$ charges, respectively.

Nontrivial commutation relations are obtained by

[TABLE]

where we define the basic electric charges as $q=-(1+\sqrt{3}\beta)/2$ and $p=-(1+\sqrt{6}\gamma-q)/3$ and the basic baryon-minus-lepton charges as $n=-(2+\sqrt{3}b)/2$ and $m=-(2+\sqrt{6}c-n)/3$. Hence, ($q,p$) and ($n,m$) will determine the $Q$ and $B-L$ charges of new particles, respectively.

II.2 Particle presentation

The fermions transform under the 3-4-1-1 gauge symmetry as

[TABLE]

where $a=1,2,3$ and $\alpha=1,2$ denote generation indices. Additionally, $\nu_{R},E,F,J$, and $K$ are new fields, included to complete the representations. This fermion content is independent of all the anomalies (cf. Appendix A).

In order for gauge symmetry breaking and mass generation, we introduce the scalar content,

[TABLE]

where the superscipts stand for ($Q,B-L$) respectively, while the subscripts indicate $SU(4)_{L}$ components. The scalars obtain such quantum numbers, provided that they couple left-handed fermions to corresponding right-handed counterparts, except that $\phi$ couples to $\nu_{R}\nu_{R}$ (see below).

II.3 Total Lagrangian

The total Lagrangian has the form,

[TABLE]

where the first part combines kinetic terms and gauge interactions, given by

[TABLE]

The covariant derivative is

[TABLE]

and we denote the coupling constants $(g_{s},g,g_{X},g_{N})$, generators $(T_{r},T_{i},X,N)$, and gauge bosons $(G_{r},A_{i},B,C)$ corresponding to the 3-4-1-1 subgroups, respectively. Above, $F$ and $S$ run over the fermion and scalar multiplets, while the parameter $\delta$ is dimensionless, called kinetic mixing.222This kinetic mixing term is always presented due to the gauge invariance and cannot be removed by rescaling the corresponding fields. Even if its tree-level value vanishes, it can be radiatively induced kineticmixing .

The second and last parts are the Yukawa interactions and scalar potential, given respectively by

[TABLE]

where the Yukawa ($h$’s) and scalar ($\lambda$’s) couplings are dimensionless, while the $\mu$’s parameters have the mass dimension.

II.4 Matter parity

Since $Q$ is conserved, only the neutral components $\eta_{1},\rho_{2},\chi_{3},\Xi_{4}$, and $\phi$ develop vacuum expectation values (VEVs),

[TABLE]

The VEVs $V,w,u,v$ break the 3-4-1-1 symmetry to $SU(3)_{C}\otimes U(1)_{Q}\otimes U(1)_{B-L}$, while the VEV $\Lambda$ breaks $B-L$ to the matter parity, $U(1)_{B-L}\rightarrow P$, where

[TABLE]

is multiplied by the spin parity $(-1)^{2s}$ as conserved by the Lorentz symmetry, similar to the 3-3-1-1 model 3311 ; dong2015 .333This kind of the matter parity is also recognized in the class of the left-right extensions 3331 .

Because $w,V,\Lambda$ provide the masses of new particles, whereas $u,v$ do so for the ordinary particles, we assume $u,v\ll w,V,\Lambda$, to keep a consistency with the standard model.

The matter parity $P$ divides particles into two types:

Normal particles according to $P=1$ (even): $\nu$, $e$, $u$, $d$, $\eta_{1,2}$, $\rho_{1,2}$, $\chi_{3}$, $\Xi_{4}$, $\phi$, $\gamma$, $W$, $Z_{1,2,3,4}$, which include the standard model particles. 2. 2.

Wrong particles according to $P=P^{\pm}_{n}\equiv(-1)^{\pm(3n+1)}$ or $P=P^{\pm}_{m}\equiv(-1)^{\pm(3m+1)}$: $E$, $F$, $J$, $K$, $\eta_{3,4}$, $\rho_{3,4}$, $\chi_{1,2,4}$, $\Xi_{1,2,3}$, $W_{13}^{\mp q,\mp(n+1)}$, $W_{14}^{\mp p,\mp(m+1)}$, $W_{23}^{\mp(q+1),\mp(n+1)}$, $W_{24}^{\mp(p+1),\mp(m+1)}$, where the $W$’s fields are non-Hermitian gauge bosons which couple to the mentioned weight-raising/lowering operators. The remainders $\chi_{4}$, $\Xi_{3}$, and $W_{34}^{\pm(q-p),\pm(n-m)}$ have $P=P^{+}_{n}P^{-}_{m}$ or conjugated.

Generally, the wrong fields transform nontrivially under the matter parity for $n,m\neq(2k-1)/3$ and $n-m\neq 2k/3$ for every $k$ integer. However, an alternative case is that both $P_{n,m}=-1$ are odd, i.e. $n,m=2k/3$. In this case all the wrong fields are odd, except that $\chi_{4}$, $\Xi_{3}$, and $W_{34}$ are even which belong to the first type of normal particles.

II.5 Dark matter

It is easily to prove that the wrong particles always couple in pairs or self-interacted due to the matter parity conservation, which is analogous to superparticles in supersymmetry dong2015 (see also Dong and Huong in 3331 ). Hence, the lightest wrong particle (LWP) is stabilized, responsible for dark matter.

Since the candidate must be color and electrically neutral, we have several dark matter models: (i) $q=0$ including $E^{0}$, $\eta^{0}_{3}$, $\chi^{0}_{1}$, $W^{0}_{13}$; (ii) $p=0$ including $F^{0}$, $\eta^{0}_{4}$, $\Xi^{0}_{1}$, $W^{0}_{14}$; (iii) $q=-1$ consisting of $\rho^{0}_{3}$, $\chi^{0}_{2}$, $W^{0}_{23}$; (iv) $p=-1$ consisting of $\rho^{0}_{4}$, $\chi^{0}_{2}$, $W^{0}_{24}$. In each case, the remaining basic electric charge is left arbitrary.

The specific dark matter models that combine above cases are

$q=p=-1$: The candidate is a scalar combination of $\rho_{3,4}$, $\chi_{2}$, and $\Xi_{2}$, or a gauge boson combination of $W_{23}$ and $W_{24}$. 2. 2.

$q=-1,p=0$: The candidate is a fermion combination of $F_{1,2,3}$, a scalar combination of $\eta_{4}$, $\rho_{3}$, $\chi_{2}$, and $\Xi_{1}$, or a gauge boson combination of $W_{14}$ and $W_{23}$. 3. 3.

$q=0,p=-1$: The candidate is a fermion combination of $E_{1,2,3}$, a scalar combination of $\eta_{3}$, $\rho_{4}$, $\chi_{1}$, and $\Xi_{2}$, or a gauge boson combination of $W_{13}$ and $W_{24}$. 4. 4.

$q=p=0$: The candidate is a fermion combination of $E_{1,2,3}$ and $F_{1,2,3}$, a scalar combination of $\eta_{3,4}$, $\chi_{1}$, and $\Xi_{1}$, or a gauge boson combination of $W_{13}$ and $W_{14}$.

The last model is for $p=q\neq 0,-1$. The candidate includes a scalar combination of $\chi^{0}_{4}$ and $\Xi^{0}_{3}$, or a vector $W^{0}_{34}$.

II.6 Fermion mass

When the scalars develop VEVs, the fermions gain masses and we write Dirac masses as $-\bar{f}_{L}m_{f}f_{R}+H.c.$ and Majorana masses as $-\frac{1}{2}\bar{f}^{c}_{L,R}m^{L,R}_{f}f_{L,R}+H.c.$

The mass matrices of new fermions $E_{a}$, $F_{a}$, $J_{a}$, and $K_{a}$ are given by

[TABLE]

which all have masses at $w$, $V$ scale.

The mass matrices of charged-leptons and quarks $e_{a}$, $u_{a}$ and $d_{a}$ are obtained as

[TABLE]

which provide appropriate masses at $u,v$ scale.

For the neutrinos, $\nu_{aL,R}$, the Dirac and Majorana masses are $[m_{\nu}]_{ab}=-h^{\nu}_{ab}\frac{u}{\sqrt{2}}$ and $[m^{R}_{\nu}]_{ab}=-\sqrt{2}h^{\prime\nu}_{ab}\Lambda$, respectively. Since $u\ll\Lambda$, the observed neutrinos $(\sim\nu_{aL})$ achieve masses via the type I seesaw mechanism,

[TABLE]

which is small, as expected. The sterile neutrinos $(\sim\nu_{aR})$ obtain large masses, such as $m^{R}_{\nu}$.

III Kinetic mixing

III.1 Canonical basis

Let us write down the kinetic terms of the two $U(1)$ gauge fields as

[TABLE]

where $B_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}$ and $C_{\mu\nu}=\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu}$ are the corresponding field strength tensors.

Because of the kinetic mixing term ($\delta$), the two gauge bosons $B_{\mu}$ and $C_{\mu}$ are generally not orthonormalized. We change to the canonical basis by a nonunitary transformation $(B_{\mu},C_{\mu})\rightarrow(B^{\prime}_{\mu},C^{\prime}_{\mu})$, where

[TABLE]

We substitute $B,C$ in terms of $B^{\prime},C^{\prime}$ into the covariant derivative. It becomes

[TABLE]

which is given in terms of the orthonormalized (canonical) fields $(B^{\prime}_{\mu},C^{\prime}_{\mu})$.

III.2 Gauge boson mass

The 3-4-1-1 symmetry breaking leads to mixings among $A_{3}$, $A_{8}$, $A_{15}$, $B^{\prime}$, and $C^{\prime}$. Their mass Lagrangian arises from $\sum_{S}(D_{\mu}\langle S\rangle)^{\dagger}(D^{\mu}\langle S\rangle)$, such that

[TABLE]

where the mass matrix $M^{2}=\{m^{2}_{ij}\}$ is symmetric, possessing the elements,

[TABLE]

where we have defined $t_{X}=g_{X}/g$, $t_{N}=g_{N}/g$, and $\beta_{1}=\sqrt{6}+\sqrt{2}\beta+\gamma$.

The mass matrix always provides a zero eigenvalue with corresponding eigenstate (photon field),

[TABLE]

where $s_{W}=e/g=t_{X}/\sqrt{1+(1+\beta^{2}+\gamma^{2})t_{X}^{2}}$ is the sine of the Weinberg’s angle donglong . Since the field in parentheses of (78) is properly the hypercharge field coupled to $Y=Q-T_{3}$, we define the standard model $Z$ as

[TABLE]

The new neutral gauge bosons, called $Z^{\prime}_{2},Z^{\prime}_{3}$, orthogonal to the hypercharge field take the forms,

[TABLE]

At this stage, $C^{\prime}$ is always orthogonal to $A,Z,Z^{\prime}_{2},Z^{\prime}_{3}$.

Let us change to the new basis $A,Z,Z^{\prime}_{2},Z^{\prime}_{3}$, and $C^{\prime}$, such that $(A_{3}\,A_{8}\,A_{15}\,B^{\prime}\,C^{\prime})^{T}=U_{1}(A\,Z\,Z^{\prime}_{2}\,Z^{\prime}_{3}\,C^{\prime})^{T}$, where

[TABLE]

The mass matrix $M^{2}$ is correspondingly changed to

[TABLE]

where

[TABLE]

where we have defined $\beta_{2}=1+(\sqrt{3}\beta+\beta^{2}+\gamma^{2})t_{X}^{2}$.

Since $u,v\ll w,V,\Lambda$, the first row and first column of $M^{\prime 2}_{s}$ consist of the elements much smaller than those of the remaining entries. We diagonalize $M^{\prime 2}_{s}$ using the seesaw formula seesaw that separates $Z$ from the heavy fields, given by

[TABLE]

where $Z_{1}$ is physical as decoupled, while $\mathcal{Z}^{\prime}_{2}$, $\mathcal{Z}^{\prime}_{3}$ and $\mathcal{C}^{\prime}$ mix via $M^{\prime\prime 2}_{s}$, such that

[TABLE]

We further separate $\epsilon_{1,2,3}\equiv\epsilon_{1,2,3}^{0}+\epsilon_{1,2,3}^{\delta}$, where $\epsilon^{0}_{1,2,3}$ are the mixing of $Z$ with $Z^{\prime}_{2},Z^{\prime}_{3},$ and $C^{\prime}$ due to the symmetry breaking, whereas $\epsilon^{\delta}_{1,2,3}$ determine those mixings due to the kinetic mixing,

[TABLE]

Because $\epsilon_{1,2,3}\sim(u^{2},v^{2})/(w^{2},V^{2},\Lambda^{2})$, the mixings are very small.

Next, the symmetry breaking is done through three possible ways, corresponding to the assumptions: $w,V\ll\Lambda$, $w\ll V,\Lambda$, or $w,\Lambda\ll V$. Let us consider the first case, $w,V\ll\Lambda$. We have the element $m^{2}_{C^{\prime}}$ much larger than the remainders. The mass matrix $M^{\prime\prime 2}_{s}$ can be diagonalized by using the seesaw formula, which yields

[TABLE]

where $Z_{4}$ is physical as decoupled, while $\mathcal{Z}_{2},\mathcal{Z}_{3}$ mix via $M^{2}_{2\times 2}$. We obtain

[TABLE]

where $\zeta_{1,2}\equiv\zeta_{1,2}^{0}+\zeta_{1,2}^{\delta}$,

[TABLE]

which are very small, and

[TABLE]

Last, we diagonalize $M^{2}_{2\times 2}$ to yield two remaining physical gauge bosons,

[TABLE]

The $\mathcal{Z}_{2}-\mathcal{Z}_{3}$ mixing angle and $Z_{2},Z_{3}$ masses are given by

[TABLE]

Now we consider two other cases, $w\ll V,\Lambda$ and $w,\Lambda\ll V$. Because $m^{2}_{Z^{\prime}_{2}}$, $m^{2}_{Z^{\prime}_{2}Z^{\prime}_{3}}$, $m^{2}_{Z^{\prime}_{2}C^{\prime}}\ll$ $m^{2}_{Z^{\prime}_{3}}$, $m^{2}_{Z^{\prime}_{3}C^{\prime}}$, $m^{2}_{C^{\prime}}$, the mass matrix $M^{\prime\prime 2}_{s}$ can be diagonalized, obeying

[TABLE]

where $Z_{2}$ is physical as decoupled, while $\mathcal{Z}_{3}$ and $\mathcal{C}$ mix via $M^{\prime 2}_{2\times 2}$, and

[TABLE]

Further for the case $w\ll V,\Lambda$, we achieve $\mathcal{E}_{1,2}\equiv\mathcal{E}^{0}_{1,2}+\mathcal{E}^{\delta}_{1,2}$, where

[TABLE]

which are very small. Otherwise, for the case $w,\Lambda\ll V$, we have

[TABLE]

which may be large.

We diagonalize the mass matrix $M^{\prime 2}_{2\times 2}$ to get two remaining physical gauge bosons, such that

[TABLE]

The $\mathcal{Z}_{3}-\mathcal{C}$ mixing angle for the case $w\ll V,\Lambda$ is given by

[TABLE]

which may be large. For the case $w,\Lambda\ll V$, the $\mathcal{Z}_{3}-\mathcal{C}$ mixing angle is defined similarly to (150), but the term associated to $\Lambda$ should be omitted. In particular, all the two cases imply $\xi=0$ when $\delta=ct_{N}/\gamma t_{X}$, the condition by which the kinetic mixing and symmetry breaking effects cancels out. Besides, the $Z_{3},Z_{4}$ masses are given by

[TABLE]

In summary, the original fields are related to the mass eigenstates by $(A_{3}\,A_{8}\,A_{15}\,B\,C)^{T}=U(A\,Z_{1}\,Z_{2}\,Z_{3}\,Z_{4})^{T}$. For the first case, $w,V\ll\Lambda$, we have $U=U_{\delta}U_{1}U_{2}U_{3}U_{\varphi}\simeq U_{\delta}U_{1}U_{2}U_{\varphi}$. For the second case, $w\ll V,\Lambda$, we obtain $U=U_{\delta}U_{1}U_{2}U^{\prime}_{3}U_{\xi}\simeq U_{\delta}U_{1}U_{2}U_{\xi}$. For the last case, $w,\Lambda\ll V$, the mixing matrix is $U=U_{\delta}U_{1}U_{2}U^{\prime}_{3}U_{\xi}$. Here we define

[TABLE]

The fields $A,Z_{1}$ are identical to the standard model, whereas $Z_{2},Z_{3}$ and $Z_{4}$ are new, heavy gauge bosons. The mixings of the standard model gauge bosons with the new gauge bosons are very small, while the mixing within the new gauge bosons may be large.

IV Electroweak precision test

IV.1 $\rho$ parameter

The new physics that contributes to the $\rho$-parameter starts from the tree-level. This is caused by the mixing of the $Z$ boson with the new neutral gauge bosons. We evaluate

[TABLE]

where

[TABLE]

This tree-level contribution is appropriately suppressed due to $u,v\ll w,V,\Lambda$. The $\rho$ deviation may receive one-loop corrections by non-degenerate vector multiplets, such as $(W_{13},W_{23})$ and $(W_{14},W_{24},W_{34})$, similar to the 3-3-1 model sturho . However, this source can be neglected if the new gauge bosons are heavy at TeV. In this analysis, we consider only the tree-level contribution.

Let us note that $\beta^{2}+\gamma^{2}<1/s^{2}_{W}-1\simeq 3.329$, which fixes $|\gamma|,|\beta|<1.82456$. The condition $q=-(1+\sqrt{3}\beta)/2$ leads to $-2.08012<q<1.08012$. Considering $q$ to be integer implies $q=-2,-1,0,$ and $1$. When $q=-2$, i.e. $\beta=\sqrt{3}$, we obtain $|\gamma|<0.57359$. The condition $p=-(1+\sqrt{6}\gamma-q)/3$ provides $-1.46833<p<-0.53167$, thus $p=-1$, i.e. $\gamma=0$, given that $p$ is integer. When $q=-1$, i.e. $\beta=1/\sqrt{3}$, we get $p=-2,-1,0$, thus $\gamma=4/\sqrt{6},1/\sqrt{6},-\sqrt{2}/\sqrt{3}$, respectively. When $q=0$, i.e. $\beta=-1/\sqrt{3}$, we gain $p=-1,0,1$, thus $\gamma=\sqrt{2}/\sqrt{3},-1/\sqrt{6},-4/\sqrt{6}$, respectively. When $q=1$, i.e. $\beta=-\sqrt{3}$, we have $p=0$, thus $\gamma=0$.

However, we are interested in the four models for dark matter, such that $q=p=-1$ (or $\beta=1/\sqrt{3},\gamma=1/\sqrt{6}$), $q=-1,p=0$ (or $\beta=1/\sqrt{3},\gamma=-\sqrt{2}/\sqrt{3}$), $q=0,p=-1$ (or $\beta=-1/\sqrt{3},\gamma=\sqrt{2}/\sqrt{3}$), and $q=p=0$ (or $\beta=-1/\sqrt{3},\gamma=-1/\sqrt{6}$).444In these cases, the Landau pole is high enough, such that the new physics is viable landau . Besides, we take $n=m=0$, thus $b=-2/\sqrt{3}$, $c=-\sqrt{2}/\sqrt{3}$, for brevity. This case implies that the wrong particles are old. On the other hand, the $W$ mass, $m_{W}^{2}=\frac{g^{2}}{4}(u^{2}+v^{2})$, implies $u^{2}+v^{2}=(246\ \mathrm{GeV})^{2}$. We will take $u$ in the range $(0,246)\ \mathrm{GeV}$, while $v$ is related to $u$.

The $\rho$ deviation is given from the global fit by $0.0002<\Delta\rho<0.00058$ pdg2018 . For the cases, $w,V\ll\Lambda$ and $w\ll V,\Lambda$, $\Delta\rho$ is independent of $\delta$. Additionally, in the latter case ($w\ll V,\Lambda$), $\Delta\rho$ is independent of $\gamma$. However, in the case $w,\Lambda\ll V$, all the parameters contribute to $\Delta\rho$, except for $V$. Without loss of generality, we impose $V=2w$ for the case $w,V\ll\Lambda$ while $\Lambda=2w$ for the case $w,\Lambda\ll V$. Besides, we put $t_{N}=0.5$.

In Fig. 1, we make a contour of $\Delta\rho$ as the function of ($u,w$) concerning the first case of VEV arrangement. Here, the panels arranging from left to right correspond to the four dark matter models such as ($\beta=1/\sqrt{3},\gamma=1/\sqrt{6}$), ($\beta=1/\sqrt{3},\gamma=-\sqrt{2}/\sqrt{3}$), ($\beta=-1/\sqrt{3},\gamma=\sqrt{2}/\sqrt{3}$), and ($\beta=-1/\sqrt{3},\gamma=-1/\sqrt{6}$), respectively.

In Fig. 2, we make a contour of $\Delta\rho$ as the function of ($u,w$) for the second case of VEV arrangement. Here, we have only two viable cases, the left panel for $\beta=1/\sqrt{3}$ and the right panel for $\beta=-1/\sqrt{3}$.

The third case depending on the kinetic mixing parameter is given in Figs. 3, 4, 5, and 6 according to the dark matter models ($\beta=1/\sqrt{3},\gamma=1/\sqrt{6}$), ($\beta=1/\sqrt{3},\gamma=-\sqrt{2}/\sqrt{3}$), ($\beta=-1/\sqrt{3},\gamma=\sqrt{2}/\sqrt{3}$), and ($\beta=-1/\sqrt{3},\gamma=-1/\sqrt{6}$), respectively. It is clear that the new physics scale bound is increased, when $|\delta|$ increases. The effect of $\delta$ is strong, when $u$ reaches values near $145$ GeV for the first dark matter model. By contrast, when $u$ approaches [math] or $246$ GeV, the effect is negligible. In summary, the kinetic mixing effect is important when the new physics is considered.

IV.2 $Z_{1}\bar{f}f$ couplings

As stated, the considering model has the mixing of the $Z$ boson with the new neutral gauge bosons. From (97) and (105), we get $Z=Z_{1}+\epsilon_{1}\mathcal{Z}^{\prime}_{2}+\epsilon_{2}\mathcal{Z}^{\prime}_{3}+\epsilon_{3}\mathcal{C}^{\prime}$, $Z^{\prime}_{2}=-\epsilon_{1}Z_{1}+\mathcal{Z}^{\prime}_{2}$, $Z^{\prime}_{3}=-\epsilon_{2}Z_{1}+\mathcal{Z}^{\prime}_{3}$, and $C^{\prime}=-\epsilon_{3}Z_{1}+\mathcal{C}^{\prime}$. Hence, the couplings of $Z_{1}$ to fermions are modified by the mixing parameters $\epsilon_{1,2,3}$. Fitting the standard model precision test, the room for the mixing parameters is only $10^{-3}$ order. Hence, we impose the bound $|\epsilon_{1,2,3}|=10^{-3}$.

It is observed that in the first case ($w,V\ll\Lambda$), $\epsilon_{3}=0$ while $\epsilon_{1,2}$ are independent of $\delta,\Lambda$. In the second case ($w\ll V,\Lambda$), $\epsilon_{2,3}=0$ while $\epsilon_{1}$ is independent of $\delta,V,\Lambda$. In the last case ($w,\Lambda\ll V$), all the parameters contribute to $\epsilon_{1,2,3}$, except for $V$. Hence, we consider only the sensitivity of the new physics scales in terms of the kinetic mixing parameter for the last case. Since the effect of kinetic mixing does not depend on the $u,v$ relation, we impose $u=v=246/\sqrt{2}$ GeV and use also the previous inputs. The results are given in Fig. 7. It indicates that the new physics regime changes when $\delta$ varies.

V FCNCs

Because the fermion generations transform differently under the gauge symmetry $SU(4)_{L}\otimes U(1)_{X}\otimes U(1)_{N}$, the tree-level FCNCs are present. Indeed, the neutral currents arise from

[TABLE]

It is clear that the leptons and exotic quarks do not flavor-change. Furthermore, the terms of $T_{3}$, $Q$, and $B-L$ also conserve flavors. Hence, the FCNCs couple only the ordinary quarks to $T_{8,15}$, such that

[TABLE]

where $q$ is denoted either $q=(u_{1},u_{2},u_{3})$ or $q=(d_{1},d_{2},d_{3})$, $T_{8}^{q}=\frac{1}{2\sqrt{3}}\text{diag}(-1,-1,1)$, and $T_{15}^{q}=\frac{1}{2\sqrt{6}}\text{diag}(-1,-1,1)$. Changing to the mass basis, $q_{L,R}=V_{qL,qR}q^{\prime}_{L,R}$ where either $q^{\prime}=(u,c,t)$ or $q^{\prime}=d,s,b$, and $(A_{3}\,A_{8}\,A_{15}\,B\,C)^{T}=U(A\,Z_{1}\,Z_{2}\,Z_{3}\,Z_{4})$, this yields

[TABLE]

It is noted that the photon always conserves flavors, $g_{0}=0$. In the first case ($w,V\ll\Lambda$), the couplings $g_{1,2,3,4}$ are

[TABLE]

In the third case ($w,\Lambda\ll V$), the coupling $g_{1}$ is identical to (174), while

[TABLE]

In the second case ($w\ll V,\Lambda$), the couplings can be obtained from those in the third case by $\mathcal{E}_{1,2}\rightarrow 0$.

The contribution of the new physics to the meson mixing is given after integrating $Z_{1,2,3,4}$ out,

[TABLE]

where the $Z_{1}$ contribution is small and omitted.

The strongest bound comes from $B^{0}_{s}-\bar{B}^{0}_{s}$ mixing, implying pdg2018

[TABLE]

We assume the sector of up quarks to be flavor diagonal, i.e. $V_{\text{CKM}}\equiv V^{\dagger}_{uL}V_{dL}=V_{dL}$. We have $|(V^{*}_{dL})_{32}(V_{dL})_{33}|\simeq 3.9\times 10^{-2}$ pdg2018 , which leads to

[TABLE]

Our remark is that since $u,v\ll w,V,\Lambda$, the l.h.s of (183) depends only on the new physics scales, not on the weak scales.

In the first case ($w,V\ll\Lambda$), the $Z_{4}$ contribution is negligible. The l.h.s of (183) is independent of $\delta$. The other inputs given previously are used, implying the bound for $w>4.36$ TeV for all the four dark matter models.

In the second case ($w\ll V,\Lambda$), the $Z_{3,4}$ contributions are negligible. The l.h.s of (183) is independent of $\beta$, $\gamma$, and $\delta$. The bound yields $w>3.9$ TeV for all the four models.

In the third case ($w,\Lambda\ll V$), since the mixing angles $\mathcal{E}_{1,2}$ are finite, the l.h.s of (183) depends on $\beta$, $\gamma$, and $\delta$, and is depicted in Fig. 8. The figure yields that the new physics regime changes when $\delta$ varies. Furthermore, those bounds are obviously lower than that given by the two case above.

VI Collider bounds

Since the new neutral gauge bosons couple to leptons and quarks, they contribute to the Drell-Yan and dijet processes at colliders.

The LEPII searches for $e^{+}e^{-}\to\mu^{+}\mu^{-}$ happen similarly to the case of the 3-3-1-1 model, where all the new gauge bosons $Z_{2,3,4}$ mediate the process. Assuming that all the new physics scales are the same order, they are bounded in the TeV scale 3311 .

The LHC searches for dijet and dilepton final states can be studied. Using the above condition, the new physics scales are also in TeV, similarly to 331p .

VII The 3-3-1-1 model revisited

The 3-3-1-1 model is based upon the gauge symmetry $SU(3)_{C}\otimes SU(3)_{L}\otimes U(1)_{X}\otimes U(1)_{N}$. Thus it contains four neutral gauge bosons $A_{3,8}$, $B$, and $C$ according to the last three gauge groups, in which $B,C$ has a kinetic mixing term, $-(\delta/2)B_{\mu\nu}C^{\mu\nu}$. The kinetic mixing effect in the 3-3-1-1 model was explicitly studied in dong2016 . Here we present only new results beyond the previous investigation.

Changing to the canonical basis, $A_{3}$, $A_{8}$, $B^{\prime}$, and $C^{\prime}$, the corresponding mass matrix $M^{2}=\{m^{2}_{ij}\}$ is given by

[TABLE]

This result is similar to that in dong2016 , except for the last element, $m^{2}_{44}$, that differs in the coefficient of $\Lambda^{2}$. Note that $t_{X}=g_{X}/g$, $t_{N}=g_{N}/g$, $\beta$, $b$, $u$, $v$, and $w$ are those parameters belonging to the 3-3-1-1 model and in this case we have $s_{W}=e/g=t_{X}/\sqrt{1+(1+\beta^{2})t_{X}^{2}}$.

Changing to the electroweak basis, $(A_{3}\,A_{8}\,B^{\prime}\,C^{\prime})^{T}=U_{1}(A\,Z\,Z^{\prime}\,C^{\prime})^{T}$, where $C^{\prime}$ is orthogonal to $A=s_{W}A_{3}+c_{W}\left(\beta t_{W}A_{8}+\sqrt{1-\beta^{2}t^{2}_{W}}B^{\prime}\right)$, $Z=c_{W}A_{3}-s_{W}\left(\beta t_{W}A_{8}+\sqrt{1-\beta^{2}t^{2}_{W}}B^{\prime}\right)$, and $Z^{\prime}=\sqrt{1-\beta^{2}t^{2}_{W}}\left(A_{8}-\beta t_{W}B^{\prime}\right)$, thus

[TABLE]

the mass matrix $M^{2}$ changes to

[TABLE]

which has the elements as given in dong2016 , in which $m^{2}_{C^{\prime}}=m^{2}_{44}$.

The light state $Z$ can be separated by using the seesaw approximation,

[TABLE]

where

[TABLE]

We separate $\epsilon_{1,2}\equiv\epsilon_{1,2}^{0}+\epsilon_{1,2}^{\delta}$, where $\epsilon^{0}_{1,2}$ are the mixing parameters due to the symmetry breaking 3311 ; dong2015 , while $\epsilon^{\delta}_{1,2}$ determine the kinetic mixing effect,

[TABLE]

where $\epsilon_{1,2}^{\delta}$ differ from those in dong2016 .

We diagonalize $M^{2}_{2\times 2}$ to obtain mass eigenstates,

[TABLE]

in which the $\mathcal{Z}^{\prime}-\mathcal{C}^{\prime}$ mixing angle and masses are

[TABLE]

Generally, $\xi$ is finite if $w\sim\Lambda$. The kinetic mixing and symmetry breaking effects cancel out if $\delta=bt_{N}/\beta t_{X}$, which takes place between $\delta$ and $b/\beta$—the embedding coefficients of $T_{8}$. Whereas, in the 3-4-1-1 model, it happens between $\delta$ and $c/\gamma$—the embedding coefficients of $T_{15}$.

Hence, the gauge states are connected to the physical states by $(A_{3}\,A_{8}\,B\,C)^{T}=U_{\delta}U_{1}U_{2}U_{\xi}(A\,Z_{1}\,Z_{2}\,Z_{3})^{T}$, where

[TABLE]

The $\rho$ deviation starts from the tree-level contribution,

[TABLE]

where

[TABLE]

In this computation, we also include one-loop contributions by the gauge vector doublet $(X,Y)$, as supplied in dong2016 .

If $\Lambda\gg w$, $\Delta\rho$ does not depend on $\Lambda$, $t_{N}$, $b$, and $\delta$. If $\Lambda\sim w$, all the parameters modify $\Delta\rho$. Comparing to dong2016 , the difference is only expressions related to $\delta$. Hence, the first case is not investigated in this work. To finalize the result, we use the parameter values similar to those in dong2016 , namely $\Lambda=2w$, $t_{N}=0.5$, $n=0$ (thus $b=-2/\sqrt{3}$), and $q=-1,0,1$ (thus $\beta=1/\sqrt{3},-1/\sqrt{3},-\sqrt{3}$, respectively).

We make a contour of $\Delta\rho$ as the function of $(u,w)$, as depicted in Figs. 9, 10, and 11 for $\beta=-1/\sqrt{3}$, $\beta=1/\sqrt{3}$, and $\beta=-\sqrt{3}$, respectively. The effect of $\delta$ is quite similar to the 3-4-1-1 model and obviously different from dong2016 .

The new physics contribution is safe, given that $|\epsilon_{1,2}|=10^{-3}$. Without loss of generality, we impose $u=v=246/\sqrt{2}$ as well as the given values of $\Lambda=2w,t_{N},\beta,b$ are used. In Fig. 12, $\epsilon_{1,2}$ are contoured as the functions of ($w,\delta$) for $\beta=-1/\sqrt{3}$, $\beta=1/\sqrt{3}$, and $\beta=-\sqrt{3}$. It is clear that the new physics regime significantly changes when $\delta$ varies, in contradiction to dong2016 .

The meson mixing is described via the effective interaction dong2016

[TABLE]

where

[TABLE]

The $B^{0}_{s}-\bar{B}^{0}_{s}$ bound leads to dong2016

[TABLE]

When $w\ll\Lambda$, the above bound translates to $w>3.9$ TeV, independent of $\beta$, $b$, $g$, $g_{X}$, $g_{N}$, and $\delta$. When $w\sim\Lambda$, using the existing values of parameters, the bound for both scales is similar to the previous case, which is quite in agreement with the conclusion in dong2016 .

VIII Conclusion

We have proved that the 3-4-1-1 model provides dark matter candidates naturally, besides supplying small neutrino masses via the seesaw mechanism induced by the gauge symmetry breaking.

The kinetic mixing effects are evaluated, yielding the new physics scales at TeV scale, in agreement with the collision bound. The kinetic mixing and symmetry breaking effects are canceled out only in the new gauge sector and differs between the 3-4-1-1 and 3-3-1-1 models.

Similar to the 3-3-1-1 model a3311 , the 3-4-1-1 model can address the question of cosmic inflation as well as asymmetric dark and normal matter, which attracts much attention.

Acknowledgments

Appendix A Anomaly checking

The anomalies that cause troublesome include $[SU(3)_{C}]^{2}U(1)_{X}$, $[SU(3)_{C}]^{2}U(1)_{N}$, $[SU(4)_{L}]^{2}U(1)_{X}$, $[SU(4)_{L}]^{2}U(1)_{N}$, $[\text{Gravity}]^{2}U(1)_{X}$, $[\text{Gravity}]^{2}U(1)_{N}$, $[U(1)_{X}]^{2}U(1)_{N}$, $U(1)_{X}[U(1)_{N}]^{2}$, $[U(1)_{X}]^{3}$, and $[U(1)_{N}]^{3}$. Let us verify each of them.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

This again confirms that the embedding coefficients $(\beta,\gamma,b,c)$ are independent of the anomalies.