*U*(1) extension of the standard model from short-distance structure of spacetime

## Abstract

In this paper, we presented a \(\mathrm {U}(1)\) extension of the SM and the corresponding consequences, based on a more fundamental structure of the spacetime. We started fundamentally from a generally covariant theory which includes a set of the fields propagating dynamically in the fundamental spacetime and respecting for the SM gauge group. We then derived, in the effective four-dimensional spacetime, an extension of the SM with the gauge group \(\mathrm {SU}(3)_C\otimes \mathrm {SU}(2)_L\otimes \mathrm {U}(1)_Y\otimes \mathrm {U}(1)_X\). Due to the structure of the spacetime, the tiny observed neutrino masses are an unavoidable consequence in this scenario. Also, the phenomenology of the new neutral gauge boson is discussed in detail.

## 1 Introduction

The present experimental observations indicate that the spacetime is a four-dimensional manifold. However, it has been believed that the presently observed spacetime is actually not fundamental because of some reasons. For example, the gravitational interaction results in the unphysical black hole and big bang singularities at which the curvature scalars of the spacetime, densities become infinite [1]. In other words, the spacetime itself may possess more complex and deeper structures at high energy regions or short distances, at which new dynamical degrees of freedom of the spacetime should be exhibited significantly. The extension of the spacetime was first proposed by Kaluza and Klein in attempting to show that a Abelian gauge field can emerge in the four-dimension spacetime from the metric of the five-dimensional spacetime [2, 3]. It was generalized to greater dimensionality to obtain the non-Abelian gauge fields [4, 5, 6, 7, 8, 9] (also see Refs. [10, 11] for reviews). In recent years, the extension of the spacetime has attracted enormous attention because of its phenomenology aspect [12, 13, 14, 15, 16, 17, 18, 19].

Our present understanding of the elementary particles and their interactions is based on the standard model (SM) whose particle content has been completely confirmed by the LHC collaborations ATLAS and CMS [20, 21]. Its gauge symmetry group is \(\mathrm {SU}(3)_C\otimes \mathrm {SU}(2)_L\otimes \mathrm {U}(1)_Y\) corresponding to the spin-1 particles: the gluons, W/Z bosons, and photon which are responsible for mediating the strong, weak, and electromagnetism forces, respectively. An important question is whether there are additional short-range gauge forces. In particular, new short-range Abelian gauge forces are the simplest extensions of the SM. They appear in grand unified theories (GUTs) [22, 23, 24, 25, 26, 27, 28], in string theoretical models [29, 30, 31, 32], in little Higgs models [33, 34, 35], in theories with extra dimensions [17, 36, 37, 38, 39, 40], and in various extensions [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66]. (Also, the readers can see Refs. [67, 68, 69] for reviews.) The search for such gauge forces is an active area at the LHC [70, 71, 72, 73, 74, 75, 76] and future colliders such as ILC. Usually, introducing an additional Abelian gauge force is through the extension of the SM gauge symmetry group by adding an additional, fundamental \(\mathrm {U}(1)\) symmetry to the theory However, there has still the possibility that the new Abelian gauge force comes from a more complex and deeper structure of the spacetime. This is motivated by the fact that one of presently well-known four fundamental forces, the gravitational interaction, arises as a result of the geometric structure of the spacetime.

In this paper, we would like to show that an additional \(\mathrm {U}(1)\) symmetry can be emerged as a result of short-distance structure of the spacetime. The \(\mathrm {U}(1)\) extension of the SM in this paper is distinguishable from the usual \(\mathrm {U}(1)\) extensions of the SM, such as the \(\mathrm {U}(1)_{B-L}\) extension. First, the additional \(\mathrm {U}(1)\) charges of the SM fermions in this paper are different to those of the SM fermions in the usual \(\mathrm {U}(1)\) extensions of the SM, because here the right-handed neutrinos do not carry the charges under this Abelian symmetry. [Whereas, in the usual \(\mathrm {U}(1)\) extensions of the SM, the right-handed neutrinos need to carry the additional \(\mathrm {U}(1)\) charges. This is necessary to generate the heavy Majorana masses for the right-handed neutrinos after the spontaneous breaking.] Second, the gauge boson associated with the additional \(\mathrm {U}(1)\) has both usual and unusual couplings. With respect to the unusual coupling, more specifically, this gauge boson can couple to a particle and an anti-particle of another field. Also, it is interesting that the small masses of the neutrinos are an unavoidable consequence of the short-distance structure of the spacetime, without imposing the Yukawa couplings between the right-handed neutrinos and exotic Higgs to get heavy Majorana masses like the usual \(\mathrm {U}(1)\) extensions. In addition, the atomic parity violation in Cesium (\(^{133}_{\phantom {k} 55}\text {Cs}\)) constrains very strongly on the spontaneous symmetry breaking scale of the additional \(\mathrm {U}(1)\), with a lower bound to be about 7 TeV. Thus, it is testable at current and future colliders.

This paper is organized as follows. In Sect. 2, we propose a more fundamental structure for the spacetime and discuss relevant important properties. In Sect. 3, we build a realistic model at which we present the particle content, the spontaneous symmetry breaking, the mass spectrum, and phenomenology of the new gauge boson. Finally, we devote to conclusion in the last section. In this work, we use units in \(\hbar =c=1\) and the signature of the metric \((+,-,-,-,...)\).

## 2 Setups

In this section, we would like to propose a more fundamental structure for the spacetime, which is expected to exhibit at (very) short distances or high energies. Then, we discuss relevant important properties, based on which in next section we will build a realistic model with the gauge symmetry group \(\mathrm {SU}(3)_C\otimes \mathrm {SU}(2)_L\otimes \mathrm {U}(1)_Y\).

### 2.1 Fundamental spacetime

^{1}By definition, the spacetime \(M_5\) consists of the following elements [78]:

- 1.
A surjective projection \(\pi : M_5\longrightarrow M_4\). Each inverse image \(\pi ^{-1}(x)\cong \mathrm {U}(1)\) is a submanifold of \(M_5\) and called a fiber.

- 2.
In general, the spacetime \(M_5\) is covered by a set of charts \(\{(V_i\times \mathrm {U}(1),\phi _i)\}\), where \(V_i\) is a local region (open subset) of \(M_4\) and \(\phi _i\) is a diffeomorphism map as \(\phi _i:\ \ V_i\times \mathrm {U}(1)\longrightarrow \pi ^{-1}(V_i)\) [where \(\pi ^{-1}(V_i)\) is of course a local region of the spacetime \(M_5\)].

*p*refers to a point in the spacetime \(M_5\). This leads to the following general coordinate transformation

*x*-coordinate change because the base manifold of the spacetime \(M_5\) is flat.] The theory is covariant with respect to this general coordinate transformation.

*active*action of the Lie group \(\mathrm {U}(1)\) on it (which turns a point of \(M_5\) into another point of \(M_5\), not to change the local coordinates of the same point), defined as [78]

*p*and

*pa*, respectively, \(\Phi \big |_{(x,e^{i\theta })}\) refers to the value of the field \(\Phi \) at the point

*p*of the local coordinates \((x,e^{i\theta })\), and

*T*[

*a*] is a representation of the element \(a=e^{i\beta }\), which is of course dependent on the field \(\Phi \), given as

*not*match the value of the field \(\Phi \) at the point

*pa*of the local coordinates \((x,e^{i\theta }a)\), meaning that

*pa*, we have

*a*, then the field \(\Phi \) is called to be invariant under the active action of the Lie group \(\text {U}(1)\). It is clearly that (6) and (8) [holding for arbitrary \((x,e^{i\theta })\) and

*a*] imply a specific function form for the field \(\Phi \) which is invariant under the active action of the Lie group \(\text {U}(1)\), as

*x*-dependence of \(\Phi \). In this way, with respect to a field which is invariant under the active action of the Lie group \(\text {U}(1)\), it itself has a specific \(\theta \)-dependence. Under the general coordinate transformation (2), we have \(\Phi '(x',e^{i\theta '})=\Phi (x,e^{i\theta })\), thus the

*x*-dependence of \(\Phi \) must transform as

### 2.2 Gauge field from the structure of spacetime \(M_5\)

In this subsection, we show that the structure of the spacetime \(M_5\) leads naturally to the existence of a definite field which transforms under the general coordinate transformation (2) in the rule of the usual gauge transformation. Then, we determine how the fundamental quantities of the spacetime \(M_5\) are expressed in terms of this field. Finally, we obtain the kinetic term of this field from the action for the spacetime \(M_5\) or the Einstein-Hilbert (EH) action.

*V*of the spacetime \(M_5\) is always decomposed into two orthogonal components (as shown in Fig. 1), independently on the choice of the local coordinate system, as

*p*of the spacetime \(M_5\) is always decomposed into a direct sum of two subspaces as

*p*of the spacetime \(M_5\), respectively. It can easily see that the vertical tangent subspace is spanned by \(\frac{\partial }{\partial \theta }\equiv \partial _\theta \). This is because under the general coordinate transformation (2) \(\partial _\theta \) transforms in a covariant way as

*G*between nearby points on the spacetime \(M_5\). For

*X*and

*Y*to be any two tangent vectors of the spacetime \(M_5\), their inner product is defined by the metric

*G*as

*G*on the spacetime \(M_5\) is specified by the pair \((G_H,G_V)\). Here, \(G_H\) is the horizontal metric which defines the inner product between any vectors belonging the horizontal tangent subspace, and \(G_V\) is the vertical metric which defines the inner product between any vectors belonging the vertical tangent subspace. On the other hand, \(G_H\) defines the invariant distance element between nearby points along the horizontal directions, whereas \(G_V\) defines the invariant distance element between nearby points along the vertical directions. The horizontal metric \(G_H\) is a tensor field whose value at any point \(p\in M_5\) belongs the space \(H^*_pM_5\otimes H^*_pM_5\), which is naturally defined by

### 2.3 Fundamental fermions on spacetime \(M_5\)

*M*is the vertical mass parameter which is naturally the order of the scale \(\Lambda \), and \(\Psi ^C=C{\bar{\Psi }}^{T}\) with

*T*denoting the transpose and \(C=i\gamma ^2\gamma ^0\) to be the charge conjugation operator satisfying the relations \(C^{-1}=C^\dag =C^T=-C\) and \(C^{-1}\gamma ^\mu C=-(\gamma ^{\mu })^T\). Note that, the vertical mass parameter can be chosen to be real without loss of generality. Then, invariant action \(S[\Psi ]\) describing the propagation of the field \(\Psi \) in the spacetime \(M_5\) reads

*M*.

*non-trivially*under the local gauge symmetry such as \(\text {SU}(3)_C\times \text {SU}(2)_L\times \text {U}(1)_Y\) or carries out conserved charges such as electric charge, then this term \(S_V[\Psi ]\) should not be invariant and thus be forbidden. This means that such field \(\Psi \) itself has no the term \(S_V[\Psi ]\) determining the \(\theta \)-dependence or the dynamics of \(\Psi \) along the vertical direction in the spacetime \(M_5\). By this, such field \(\Psi \) itself must have a specific \(\theta \)-dependence which should be determined by a certain property. (Also, such field \(\Psi \) has no Kaluza-Klein excitations in the effective four-dimensional spacetime.) This property is naturally that the field \(\Psi \) is invariant under the active action of the Lie group \(\text {U}(1)\), which has been discussed in Sect. 2, meaning that

## 3 A realistic model

### 3.1 Fermion and gauge sectors

*non-trivially*under the gauge symmetry group \(\mathrm {SU}(3)_C\otimes \mathrm {SU}(2)_L\otimes \mathrm {U}(1)_Y\), the vertical kinetic term \(S_V[\Psi ]\) in the action (37) should not be invariant and thus be forbidden, except the right-handed neutrinos \(N_{aR}\). And, as a result, they themselves must have a specific \(\theta \)-dependence as

*x*is a free parameter. From (44), it is easily to see that the transforming parameters in (43) are completely independent on the fiber coordinate \(\theta \) but only dependent on the

*x*-coordinates, meaning that

*x*-coordinates and have all the zero vertical component.

The \(\mathrm {U}(1)_X\) charges of the fermions. Note that, \(X_{\nu _{aL}}=X_{e_{aL}}=X_L\), \(X_{e_{aR}}=X_E\), \(X_{u_{aL}}=X_{d_{aL}}=X_Q\), \(X_{u_{aR}}=X_U\), and \(X_{d_{aR}}=X_D\)

Fermion | \(\nu _{aL}\) | \(e_{aL}\) | \(e_{aR}\) | \(u_{aL}\) | \(d_{aL}\) | \(u_{aR}\) | \(d_{aR}\) |
---|---|---|---|---|---|---|---|

\(X_f\) | \(-x\) | \(-x\) | \(-2x\) | \(\frac{1}{3}x\) | \(\frac{1}{3}x\) | \(\frac{4}{3}x\) | \(-\frac{2}{3}x\) |

*f*(

*x*) is related to the fundamental fermion \(F(x,e^{i\theta })\) by

But, there are remarkable differences compared to the usual \(\mathrm {U}(1)\) extensions of the SM, which we will see below. Second, the fermions and the gauge bosons of the SM have no the Kaluza-Klein (KK) counterparts. This is due to the fact that the vertical kinetic term \(S_V[\Psi ]\) of the corresponding bulk fermions is forbidden because they transform non-trivially under the gauge symmetry group \(\mathrm {SU}(3)_C\otimes \mathrm {SU}(2)_L\otimes \mathrm {U}(1)_Y\).

*X*to the KK excitations of \(\nu _{aR}\) are unusual. More specifically, the gauge boson

*X*couples to a KK excitation and KK anti-excitation of another field.

### 3.2 Scalar sector and symmetry breaking

*H*and a singlet complex scalar field \(\Phi \), as

*H*as

*X*.

*X*is expressed as

*X*is lighter than all KK excitations. After the symmetry breaking, the KK excitations \(\{\phi _{1n},\phi _{2n}\}\) are still the physical fields of the mass \(\mu _n\). Whereas, the CP-even excitations of \(\phi \) and \(\varphi \), which are not eaten by the gauge bosons \(\{W^\pm , Z_\mu , X_\mu \}\), should mix together. We expand these scalar fields in the unitary gauge around the vacuum as

*h*and \(h'\) leads to the presence of the direct couplings between the gauge boson \(X_\mu \) with the SM Higgs \(h_1\), although the SM scalar doublet \(\phi \) does not carry the charge under the group \(\text {U}(1)_X\). The coupling of the exotic Higgs \(h_2\) to the SM particles generically goes through the gauge boson

*X*and the mixing between it and the SM Higgs.

### 3.3 Mass spectrum for fermions

*H*are given by the bulk Yukawa action

*x*. It is important to note here that the Dirac masses \(\left\{ m^\psi _{nab},m^\chi _{nab}\right\} \) should decrease in increasing of the KK excitation order

*n*. On the other hand, \(m^\psi _{nab}\) and \(m^\chi _{nab}\) should approach zero as \(n\rightarrow \infty \). \(\{h^e_{ab},h^u_{ab},h^d_{ab}\}\) are identified as the usual Yukawa coupling constants in the SM, thus the masses of the fermions in both charged-lepton and quark sector are the same as the SM one.

### 3.4 Phenomenology of the gauge boson *X*

*X*has the couplings with quarks and leptons. Thus, it may be produced in (future) colliders, then it would decay to pairs of fermions. Now we study the tree-level decays of the gauge boson

*X*into the two-body final states. (The loop induced decays of the gauge boson

*X*will be studied in our future work.) Because there has no tree-level mixing between the gauge boson

*X*and the SM gauge bosons,

*X*should not decay into SM diboson pairs and the pairs

*ZS*with

*S*referring to any scalar. Also, the gauge boson

*X*can not decay into the pairs of \(({\bar{\chi }}_{nR},\psi _{nR})\) and \(({\bar{\psi }}_{nR},\chi _{nR})\), because it is lighter than these KK excitations. On the other hand, in this scenario, the gauge boson

*X*may only decay into the SM fermion pairs at the tree level. The couplings of the gauge boson

*X*to the fermion

*f*is given as

*X*to the left and right handed fermions \(f_L\) and \(f_R\), respectively. Note that, with respect to the neutrinos, we have \(\lambda _{\nu _{_R}}=0\). In this scenario, because of \(\lambda _{f_L}\ne \lambda _{f_R}\), the couplings of the gauge boson

*X*to the left-handed and right-handed fermions are different. The partial decay width of the decay \(X\rightarrow {\bar{f}}f\) is given by

*f*, respectively. The invisible decay width of the gauge boson

*X*, which corresponds to its decay into neutrino and anti-neutrino pairs, is given

*X*as the functions of its mass \(M_X\), in Fig. 2, which show

*X*can be accessible via these channels.

*X*should contribute to the cross sections of \(e^+e^-\rightarrow f^+f^-\) performed at LEP-II. For \(M_X\) much larger than the collider energy \(\sqrt{s}\), it should lead to the effective four-fermion interactions induced by the exchange of the gauge boson

*X*as

*X*as

*X*should appear as a \({\bar{f}}f\) resonance and the corresponding coupling is constrained by LEP-II data [69, 79] as

*X*to the electron and the quarks

*u*and

*d*. The measurement of the nuclear weak charge of Cesium is given by [80, 81, 82, 83, 84, 85]

*Z*, \(\theta _W\) is the Weinberg angle, \(Z=55\) and \(N=78\). Then, we obtain a bound as

*X*to the anomalous magnetic moment of the muon. The deviation between the experiment and theory is now [92, 93]

*X*, the exotic Higgs \(h_2\), the neutral KK excitations \(\{\phi ^0_{1n},\phi ^0_{2n}\}\), as shown in Fig. 3. (Note that, \(\{\phi ^0_{1n},\phi ^0_{2n}\}\) are neutral components of the KK doublets \(\{\phi _{1n},\phi _{2n}\}\), respectively.)

*X*. The contribution of the gauge boson

*X*reads [94, 95]

*X*to the anomalous magnetic moment of the muon is constrained as

*X*is not enough responsible for the deviation between the experiment and the theory. On the other hand, this deviation should be explained by new physics beyond this model.

### 3.5 Discrimination and implications

If the signal of a new neutral gauge boson \(Z'\) associated with \(\text {U}(1)'\) is discovered at the accelerators, the next task will be to specify the underlying model which predicts it. Many studies of the gauge boson \(Z'\) focus the \(\text {U}(1)'\) models which occur in the decomposition of GUTs, based on the exceptional group \(\mathrm {E}_6\) and the group \(\mathrm {SO}(10)\) [23, 24, 25, 26, 27, 28]. Since, here we are interested in distinguishing our model to these models.

The \(Z'\) couplings to the SM fermions for the models generated from \(\mathrm {E}_6\) and \(\mathrm {SO}(10)\), with an overall \(e/\cos \theta _W\) factored out and \(\alpha _{LR}\simeq 1.59\)

\(\chi \) | \(\psi \) | \(\eta \) | \(\text {LR}\) | |
---|---|---|---|---|

\(q_L\) | \(-\frac{1}{2\sqrt{6}}\) | \(\frac{\sqrt{10}}{12}\) | \(\frac{1}{3}\) | \(-\frac{1}{6\alpha _{LR}}\) |

\(u_R\) | \(\frac{1}{2\sqrt{6}}\) | \(-\frac{\sqrt{10}}{12}\) | \(-\frac{1}{3}\) | \(-\frac{1}{6\alpha _{LR}}+\frac{\alpha _{LR}}{2}\) |

\(d_R\) | \(-\frac{3}{2\sqrt{6}}\) | \(-\frac{\sqrt{10}}{12}\) | \(\frac{1}{6}\) | \(-\frac{1}{6\alpha _{LR}}-\frac{\alpha _{LR}}{2}\) |

\(l_L\) | \(\frac{3}{2\sqrt{6}}\) | \(\frac{\sqrt{10}}{12}\) | \(-\frac{1}{6}\) | \(\frac{1}{2\alpha _{LR}}\) |

\(e_R\) | \(\frac{1}{2\sqrt{6}}\) | \(-\frac{\sqrt{10}}{12}\) | \(-\frac{1}{3}\) | \(-\frac{1}{2\alpha _{LR}}-\frac{\alpha _{LR}}{2}\) |

The ratio between BR of the top quark and BR of all quarks

\(\chi \) | \(\psi \) | \(\eta \) | \(\text {LR}\) | \(U(1)_X\) | |
---|---|---|---|---|---|

\(\frac{\text {BR}_{\text {top}}}{\text {BR}_{\text {total}}}\) | \(\simeq 5.55\%\) | \(\simeq 16.66\%\) | \(\simeq 20.51\%\) | \(\simeq 12.42\%\) | \(\simeq 25.82\%\) |

*X*is identified as hypercharge. With respect to the scalar sector, the scalar field given in (60) is identified as the doublet of the symmetry group \(\mathrm {SU}(2)_L\). Similarly, one can determine the relation between the gauge coupling constant \(g'\) associated with \(\text {U}(1)_Y\) and the bulk radius

*R*as, \(g'=\frac{\sqrt{2}}{M_{Pl}R}\). This implies that \(R^{-1}\) is close to the Planck energy scale or in other words the size of the fibers must be extremely small. Furthermore, we can determine the hypercharge of the fermions from the absence of the nontrivial anomalies \(\{\left[ \mathrm {SU}(3)_C\right] ^2\mathrm {U}(1)_X\), \(\left[ \mathrm {SU}(2)_L\right] ^2\mathrm {U}(1)_X\), \(\left[ \mathrm {U}(1)_X\right] ^3\), \(\left[ \text {Gravity}\right] ^2\mathrm {U}(1)_X\}\). For the lepton doublet \(l_a\), the right-handed electron \(e_{aR}\), and the quark doublet \(q_a\), their hypercharge is obtained as

*X*. Thus, this framework is possible to interpret the interaction associated with the hypercharge gauge group \(\mathrm {U}(1)_Y\) as a result of the non-trivial structure of the spacetime. For generalizing this unification for both \(\mathrm {SU}(3)_C\) and \(\mathrm {SU}(2)_L\), the fiber should be a Lie group manifold \(\mathrm {SU}(3)\otimes \mathrm {SU}(2)\otimes \mathrm {U}(1)\). However, in order to claim whether this framework provides the unification for both \(\mathrm {SU}(3)_C\) and \(\mathrm {SU}(2)_L\), we need to treat more care and detail which is beyond the scope of the present work.

*X*, there are many new particles as follows: the sterile neutrinos \(\nu '_{aR}\) and their KK partners \(\{\psi '_{naR},\chi '_{naR}\}\), the exotic Higgs \(h_2\), and the KK excitations \(\{\phi _{1n},\phi _{2n}\}\) which originate from the bulk scalar doublet

*H*. Obviously, the exotic Higgs \(h_2\) is unstable, because it is more (much) heavy than the SM Higgs \(h_1\) and thus it should decay into \(h_1\) through the coupling, \(\simeq \lambda _3v'c^3_\alpha h^2_1h_2\). In order to see that the remaining new particles are also unstable, we write the couplings, obtained from (72), as

*x*satisfies the condition (75), all the coupling constant in (108) are always different to zero. \({\mathcal {L}}_1\) describes the couplings of the sterile neutrinos \(\nu '_{aR}\) and their KK partners \(\{\psi '_{naR},\chi '_{naR}\}\) to the SM leptons and the SM Higgs \(h_1\). \({\mathcal {L}}_2\) describes the couplings of the KK excitations \(\{\phi _{1n},\phi _{2n}\}\) to the SM fermions. Finally, \({\mathcal {L}}_3\) describes the couplings which \(\{\nu '_{aR},\psi '_{naR},\chi '_{naR},\phi _{1n},\phi _{2n}\}\) couple to with together and to the SM leptons. Because of the presence of these couplings, the sterile neutrinos \(\nu '_{aR}\) and all KK excitations will decay into the SM particles. In this sense, there has no any good relic candidate in this model. However, this model may contain a good relic candidate and thus may account for dark matter, if the scalar sector of this model is extended at which we introduce additionally an inert scalar field \(\Phi '\) similar to \(\Phi \) but with \(X_{\Phi '}\ne X_{\Phi }\). Because both \(\Phi \) and \(\Phi '\) have no the Yukawa couplings to the fermions, the action of theory is accidentally invariant under the \(Z_2\) transformation

*X*and the Higgs \(h_1\). A detail study of dark matter in this model will be taken in our upcoming work.

## 4 Conclusion

In this paper, we have proposed a \(\text {U}(1)_X\) extension of the SM, which departs fundamentally from the theory in the five-dimensional fiber bundle spacetime with the SM gauge symmetry group. Here, the base manifold and fiber of the spacetime are the four-dimensional Minkowski-flat manifold and the Lie group manifold \(\mathrm {U}(1)\), respectively. The local gauge transformation under \(\text {U}(1)_X\) is originated from the general transformation in the fiber coordinate of the spacetime. The \(\text {U}(1)_X\) charges of the SM fermions are the quantum number characterizing the active action of the Lie group \(\text {U}(1)\) on the corresponding fundamental fermions. In particular, this scenario allows to explain naturally the small masses of the neutrinos as a result of the short-distance structure of the spacetime.

We have studied the tree-level decays of the new gauge boson into the two-body final states, at which its branching ratios and dominant decay channels are discussed. The bound on the \(\mathrm {U}(1)_Y\) breaking scale or the \(\mathrm {U}(1)_X\) coupling constant has been obtained from the LEP-II constraints. In particular, the atomic parity violation in Cesium (\(^{133}_{\phantom {k} 55}\text {Cs}\)) imposes a very strong constraint on the \(\mathrm {U}(1)_X\) breaking scale. Finally, we have analyzed the contribution of the new gauge boson to the anomalous magnetic moment of the muon, which requires additional contributions from new physics beyond this model.

## Footnotes

- 1.
Because the fact that the usual 4

*D*gravitation taken into account does not affect the aim of the present work, for simplicity we consider the extension of the 4*D*Minkowski-flat spacetime. In the case of that the usual 4*D*gravitation is included, the base manifold should be no longer flat but curved.

## Notes

### Acknowledgements

We would like to express sincere gratitude to the referee for his constructive comments and suggestions which have helped us to improve the quality of the paper.

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