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Eur. Phys. J. C (2019) 79:779 https://doi.org/10.1140/epjc/s10052-019-7260-z

Regular Article - Theoretical

Extension of the of electroweak and in the tangent bundle geometry

Joachim Herrmanna Max Born Institute, Max Born Straße 2a, 12489 Berlin, Germany

Received: 8 February 2019 / Accepted: 1 September 2019 / Published online: 20 September 2019 © The Author(s) 2019

Abstract A generalized theory of In this paper we consider the general hypothesis that the is developed based on the underlying geometrical structure fundamental physical can be described within of the tangent bundle with symmetries arising from trans- the geometrical structure of the most fundamental fiber formations of tangent vectors along the fiber axis at a fixed bundle—the tangent bundle, and gauge transformations can spacetime point given by the SO(3,1) group. Electroweak be identified with transformations at a fixed spacetime point interaction beyond the standard model (SM) is described by along the tangent vector axis leaving the scalar product invari- the little groups SU(2)⊗Ec(2) (Ec(2) is the central extended ant. This means the gauge group is not assumed for phe- Euclidian group) which includes the group SU(2) ⊗U(1) as nomenological reasons but is taken to arise self-consistently a limit case. In addition to isospin and , two addi- from the invariance of the scalar product with respect to tan- tional quantum numbers arise which explain the existence of gent fiber transformations described by the group SO(3, 1). families in the SM. The connection coefficients yield the SM Since the action of this group is not transitive, the vector space gauge potentials but also hypothetical gauge and other decomposes into different orbits and the most general (pro- hypothetical particles as a Higgs family as well as candidate jective) irreducible representations of SO(3,1) can be found Dark Matter particles are predicted. Several important con- by the little groups SU(2), Ec(2) and SU(1, 1) where the sequences for the interaction between dark , dark group Ec(2) is the central extended Euclidean group. Based scalars or dark vector gauge bosons with each other and with on differential geometry on the tangent bundle with covariant SM Higgs and Z-bosons are described. derivatives determined by the generators of the transforma- tion group G = SU(2) ⊗ Ec(2) and corresponding connec- tion coefficients (gauge potentials) a generalized theory of the 1 Introduction electroweak interaction is derived. In addition to the internal quantum numbers (IQN) of isospin and hypercharge, the Ec-  The formal equivalence of gauge theories with the geometry and the family quantum number n arise which could elucidate the existence of families in the SM. In this approach of fiber bundles has been recognized since the 1960s [1–5]. In ± the fiber bundle formalism, gauge potentials are understood the known Z and W gauge bosons can be found again, but c ± as a geometrical entity—the connections on the principal in addition new extra E and B gauge bosons and other bundles, and matter fields are described by associated fiber hypothetical particles as e.g. a family of Higgs particles are bundles. The geometrical interpretation of gauge theories by predicted. A notable feature of the theory presented is the pos- the mathematical fiber bundle theory is a beautiful and math- sibility of identifying candidate stable or unstable hypothet- ematically profound concept. However, in earlier investiga- ical Dark Matter (DM) vector bosons, DM scalars and DM tions the transformation groups of the fibers were taken from fermions with zero hypercharge and zero isospin but nonzero c  = the phenomenologically determined internal gauge groups E -charge 0 without additional phenomenological of the Standard Model (SM). Therefore, up to now the fiber model assumptions. Here we present only the basics, specific bundle interpretation yields mainly a re-interpretation of the in-depth observable consequences are beyond the scope of gauge fields and did not effectuated a physical theory beyond the present paper. Note that the more general transformation ( , )  ( , ) ( , ) the SM. group SO 3 1 T 3 1 (where T 3 1 is the translational group and  represents the semi-direct product) includes teleparallel into the tangent bundle geometry based a e-mail: [email protected] 123 779 Page 2 of 15 Eur. Phys. J. C (2019) 79 :779 on translational transformations T (3, 1) of the tangent fibers. not change the scalar product in (3). Under general coordi- According to this approach the interpretation of the SO(3, 1) nate transformations of the spacetime manifold xμ → yμ = connection coefficients as electroweak gauge potentials is yμ(x) vectors transform as vμ(x) = (∂yμ/∂xν)vν(x).On compatible with teleparallel gauge gravity theory, which is the other hand, the vector components in the tetrad frame fully equivalent to Einstein’s general relativity theory. basis remain unchanged: va(x) = va. A second type of transformations exists that does not change the scalar prod- uct. These are transformations at a fixed point x of the space- 2 Differential geometry on the tangent bundle time manifold M transforming the tangent vectors along the tangent fiber directions as follows: We start with a brief description of the geometry of the tan- gent bundle on a manifold (see e.g. [6]). The tangent space va = a( )vb, ´a = ( a( )) b , Tb x eμ Tb x eμ T (M) at the point x on the spacetime manifold M is the set μ − μ x e´ (x) = (T b(x)) 1e (x), (4) of all tangent vectors spanned by frame vectors in the coor- a a b dinate basis eμ = ∂μ (μ = 0, 1, 2, 3). The tangent bundle is a( ) the union of all tangent spaces at all points x of the mani- where Tb x are matrices satisfying the conditions  η a b = η fold M : TM = x∈M Tx (M). In a coordinate description a abTc Td cd. On the other hand, the tangent vectors point in TMis described by the numbers of pairs X = (x, u) which refer to the coordinate frame remain unchanged: μ μ with x ={x0, x1, x2, x3) as the coordinates of the spacetime v = v . The transformation group of tangent vectors along manifold and u ={u0, u1, u2, u3) are the coordinates of the the tangent fiber is the SO(3, 1) group of special linear trans- a( ) ∈ ( , ) tangent vectors. Thus the tangent fiber bundle geometry intro- formations, with matrix elements Tb x SO 3 1 depend- duces four additional variables u for the description of the ing on the spacetime point x as a parameter. tangent fiber. To aid understanding, it is convenient to con- Note that the transformation of the tangent vectors by the sider M as a pseudo-Riemannian spacetime manifold with group SO(3, 1) is not the most general transformation. Actu- indefinite metric gμν(x). Besides the frame vectors in the ally, the fact that the group SO(3, 1) leaves the scalar product coordinate basis eμ = ∂μ (μ = 0, 1, 2, 3), one can introduce of tangent vectors invariant is not sufficient because we need the tetrads as another geometric object on the tangent space: the infinitesimal tangent vector line elements to be invariant. We have = μ( )∂ . ea ea x μ (1) μ ν a b (dv,du) = gμν(x)dv du = ηabdv du . (5) Each vector described in the coordinate basis eμ = ∂μ can be expressed by a vector with respect to the tetrad frame basis This allows us to add constant translations to the transforma- ea according to the rule tions in (4): ν ν v = ( )va.  ea x (2) v a = a( )vb + a( ), Tb x a x (6) The subscripts a, b,...number the vectors (a, b = 0, 1, 2, 3) and this leads to the more general transformation group and μ their components in the coordinate basis. The dual a SO(3, 1)  T (3, 1). basis of the frame fields ea are cotangent frame 1-forms e = a μ μ( ) a( ) = Poincaré transformations and the transformation group (6) eμdx satisfying the orthogonality relation ea x eν x a μ of tangent vectors T (x) in the tetrad basis are described by δν . By using the tetrads of the pseudo-Riemannian mani- b the same group SO(3, 1)  T (3, 1) but the two have a prin- fold, the scalar product of two vectors is given by the Lorentz cipally different geometrical and physical meaning: the for- metric: mer transforms the spacetime coordinates of a flat manifold while the latter describes transformations within the tangent (v, ) = ( )vμ ν = ( ) μ( ) ν( )va b u gμν x u gμν x ea x e x u b fiber F = Tx (M) leaving the spacetime point x unchanged. a b = ηabv u , (3) The fact that Poincaré transformations (defined as coordi- nate transformations of spacetime in a flat manifold) and the where ηab = diag(−1, 1, 1, 1) is the metric of the Minkovski transformation (6) are based on the same mathematical group space. could lead to confusion, which can be avoided if the princi- The geometric properties of manifolds are usually related pally different meaning of the two transformations is taken to the invariance of certain geometrical structure relations into account. As an example, the Coleman–Mandula theorem under the action of certain transformation groups. The def- [7] states that the combination of spacetime symmetries with inition of the scalar product (3) is the governing structure internal symmetries is not possible in any but the trivial way. relation defining the geometry of the tangent bundle. Tan- And this is what here is the case, the group SO(3, 1) in (4) gent vectors manifest two kinds of transformations which do or SO(3, 1)  T (3, 1) in (6) are the internal groups and they 123 Eur. Phys. J. C (2019) 79 :779 Page 3 of 15 779

∂ are combined with spacetime transformations as explained = + ωa + i ab , Dμ μ i .μPa ..μMab (10) below (3) only in a trivial way. ∂x 2 where Mab are related with the six generators of the group SO(3, 1) with Ja = abcMbc, Ka =−Moa and Pa are the a 3 Connections on the tangent bundle and teleparallel generators of the translational group T (3, 1).Hereω.μ and ab gauge gravity theory ..μ are connection 1-forms of the T (3, 1) and SO(3, 1) group, respectively. The total field strength tensor can be The geometric construction of tetrads is closely linked to the defined as conceptional basis of gravity theories and its extensions to a 1 ab Fμν =[Dμ, Dν]=T.μν Pa + R..μν Mab, (11) gravity gauge theories. To facilitate a proper understanding 2 of the underlying geometric structure and a unified descrip- with the torsion tensor tion including gravity we first consider the general inhomo- a = ∂ ωa − ∂ ωa + ( a ωe − a ωe ), geneous transformation group SO(3, 1)  T (3, 1) and its T.νμ ν .μ μ .ν .eν .μ .eμ .ν (12) relationship with gravity. The differential geometry on the and the curvature tensor tangent bundle can be obtained using the general rules for a = ∂ a − ∂ a + ( a e − a e ). principal fiber bundles P(M; G) requiring the definition of R.bνμ ν .bμ μ .bν .eν .bμ .eμ .bν (13) connections and covariant derivatives on the bundle. The def- Through contraction with tetrads, tensors can be transformed inition of a covariant derivative demands considering vectors μ μ a to spacetime indexed forms as e.g. v = ea v , R.λνμ = which point from one fiber to the other at different points x e eb Ra and the lower frame index v can be raised by and x´ of the spacetime manifold. The generators L of the a λ .bνμ a a va = ηabv ωa group G are vertical vectors pointing along the fiber and the Lorentz metric b. The connections .μ and ab are fundamental structure functions characterizing the therefore belong to the vertical subspace Vu(P). Horizontal ..μ specific tangent bundle. vectors in the subspace Hu(P) which point away from the va fibers (i.e. elements of the tangent space of the fiber bun- Parallel transport of a tangent vector from a point x in the spacetime manifold to a neighboring point x´ is defined dle Tu(P) that complement the vertical vectors in Vu(P)) can be constructed by the definition of a connection as an by the covariant derivative assignment to each point in the principal fiber such that [8,9] va = ∂ va + a vb, Dμ μ .μb (14) T (P) = H (P) ⊕ V (P). a u u u (7) where .μb is called a frame connection. On the other hand, the covariant derivative of vectors which refer to the coordi- The definition of a connection can be used for the definition nate basis can be written as of a covariant differentiation along the curves horizontally ν ν ν ρ lifted to the principal bundle: Dμv = ∂μv + .μρv , (15) μ ν d dx with the coordinate connection .μρ. The coordinate con- = Dμ, ρ τ τ (8) a d d nection .μλ is connected with the frame connection .bμ a = ∂ a − ρ a + a b = where by requiring Dμeν μeν .μνe .μbeν 0from ∂ which the following relation can be derived: = + a Dμ μ i AμLa (9) a = a∂ ν + a ν . ∂x .bμ eν μeb eν .ρμeb (16) is the covariant derivative on the principal fiber bundle. The For the description of gravity tetrads eν and a specific coor- ρ b La are the right-invariant fundamental vector fields (genera- dinate connection .μλ has to be defined. For a manifold a tors) on the group manifold G ={gij} and Aμ the connec- with vanishing metricity described by the condition Dλgμν = ρ tion coefficients of the group G. A connection on a principal ∂λgμν − .μλg ν − .νλgμρ = 0 one gets [10] bundle induces a connection on the associated bundle. The ρ =  + , covariant derivative on the associated bundle is given by (9) .μν .μν K.μν (17) substituting the generators L by the left-invariant funda- ρ a where .μν is the Levi-Civita connection and K.μν is the mental vector fields on the section of the associated bundle, contortion tensor, which describe matter fields. 1 . . The geometric transformations of tangent vectors in a tan- K.μν = (Tν.μ + Tμ.ν − T.μν), (18) gent bundle are described by the group G = SO(3, 1)  2

T (3, 1). According to (9) the covariant derivative along the with the torsion tensor T.μν = .μν − .νμ. The under- horizontal lifted curve on the principal bundle P(M; G) of lying geometric structure with respect to translational and this group is given by rotational transformations (6) of the tangent vectors leads to 123 779 Page 4 of 15 Eur. Phys. J. C (2019) 79 :779

= ( a ) = a a Riemann–Cartan spacetime endowed with frame connec- where h det eμ and Tb Tab. The Lagrangian is up to a ωa a a tions .ν and .μb, nonvanishing curvature R.bνμ, nonvan- divergence equivalent to the Lagrangian of Einstein’s GTR, a ishing torsion T.νμ but vanishing metricity. The special case this means the two theories are simply alternative formula- of a Riemann spacetime and the General Theory of Relativity tions for the description of gravity but are based on different (GTR) can be obtained from the above formulas by setting principles. the torsion tensor to be identically vanishing. The coordinate A principal bundle P(M; G) encodes the essential data ν a connection .ρμ then is given by the Levi-Civita connection of gauge transformations and the frame connection .bν and  a .μν, related with the metric tensor gμν. The frame connec- the ω.μ are additional structure functions that are attached to a a it and are in general independent defined on the existence of a tion .μb in this case is usually called a spin connection .bν;  = ( , ) ( , ) it is related with the Levi-Civita connection .μν by Eq. (16). metric ([8,9]). If the structure group G SO 3 1 T 3 1 ( , ) On the other hand, based on the with respect of is restricted to the translational subgroup T 3 1 Eqs. (12) a translational transformations of tangent vectors a gauge the- and (13) can be taken with .bμ everywhere put equal a = ory of gravity has been developed by analogy with the internal to zero. The torsion tensor is now determined by T.νμ ∂ ωa − ∂ ωa symmetries [11–15]. This gauge gravity theory corresponds ν .μ μ .ν, which can be obtained by a specific gauge to the teleparallel gravity theory, which is an alternative but fixing and describes gravitation in the so-called pure telepar- equivalent formulation of the GTR describing the very same allel gravity theory. Note that the structure group of the tan- = ( , )  ( , ) gravitational field. In teleparallel gravity, the torsion is non- gent bundle is the larger group G SO 3 1 T 3 1 . vanishing, acting as a gravitational while the curvature This raises the question of the physical meaning of the other a a ωa subgroup SO(3, 1).Ifthe . μ do not vanish we find from R.bνμ vanishes identically. The translational connections μ b a and the tetrad coframe eμ turn out to be conceptional dis- (13) a non-Abelian field strength tensor which in teleparallel tinct entities, since it does not transform inhomogeneously gauge gravity theory (based on the translational symmetry) with a gradient term under a gauge transformation [18]. The is not related to gravity. In this paper the main hypothesis coframes depend on the translational connection in the form is elaborated that non-Abelian fields in electroweak interac- a a = ωa + ξ a = ∂ δ + a ξ a = ξ a( ) tion can be identified with the connection coefficients . μ eμ μ Dμ , Dμ μ ab .μb where x is a b coset vector. Locally at a given point x on the spacetime man- arising from transformations along the tangent fiber axis ( , ) ifold one can transform Dμξ a by a gauge tansformation into described by the group SO 3 1 . This interpretation differs from Poincaré gravity based on the localization δaμ (where δ is the Kronecker symbol). Note that the coframe a eμ induces a metric as an independent dynamically quantity. of the Poincaré group as gauge group [16] (for a review see In the geometry of teleparallel gravity the coordinate con- [17]). In this theory both the translational part and the rota- ρ tional part of the local Poincaré group is related with gravity nection .μλ takes the form of the Weizenböck connection  leading to a hypothetical generalized gravity theory, denoted ρ =W ρ ( ), .μλ .μν x defined as as the Einstein–Cartan gravity theory. In the following we use the terms connection coefficients and gauge potentials in W ρ a .μν(x) = ea (x)∂νeμ(x). (19) parallel as well as gauge transformations and tangent vector transformations. If we consider only gravitational effects with the choice of the Weizenböck connection in teleparallel gravity a vector 4 The generators on the little groups is parallel transported if its projection on the tetrads is pro- portional, regardless of the path connecting the two tangent From now on we neglect gravity arising from the transla- spaces. This can we see from (15) with the substitution of ρ ρ tional part aa(x) of the transformation of tangent vectors in .μν(x) by W .μν(x), which yields for the covariant deriva-  ν ν a (6). Since the action of SO(3, 1) on a tangent vector is not tive Dμv = e (x)∂μv . In teleparallel gravity the connec- a transitive, the vector space decomposes into different orbits a tion coefficients .bν represents a pure inertial effect [14] ( ), ( )  with the little groups SO 3 E 2 and SO(1,2). The uni- where a is the spin connection in teleparallel gravity. .bν tary representations TL (g) of the little groups SO(3) and a There exist a class of inertial frames in which .bν vanishes: E(2) are well known. The composition law of these so-called a = ( ) .bν 0. vector representations TL g satisfy the functional equation The Weizenböck torsion can be used to build up the TL (g1)TL (g2) = TL (g1g2) and encode the law of group Lagrangian of teleparallel gravity with quadratic Weizen- transformations on the set of vector states. However, it is böck scalars [11,14,15], given by well known that this composite law is too restrictive and   leads in special cases to certain pathologies, as e.g. the Dirac h 1 abc 1 abc a equation is not invariant under the Poincaré group, but under L =− T Tabc + T Tbac − T Ta , (20) 16π 4 2 its universal covering group. In quantum theory the physical 123 Eur. Phys. J. C (2019) 79 :779 Page 5 of 15 779 symmetry of a group of transformations on a set of vec- where m(α, a,ξ1,ξ2) is the two-cocycle which gives the tor states has to preserve the transition probability between desired central extension parametrized as two vector states |≺ , T (g) |2 = |≺ ,  |2. There- L m(α, a,ξ ,ξ ) = (a ξ + a ξ ) sin α fore, as shown by Wigner [19] and systematically studied 1 2 1 1 2 2 −( ξ − ξ ) α. by Bargman [20], the problem of pathologies can be solved a1 2 a2 1 cos (24) if the composite law given above is replaced by a weaker By using (23) and (24) we can find the infinitesimal transfor- ( ) ( ) = ε( , ) ( ) ε( , ) one: TL g1 TL g2 g1 g2 TL g1g2 where g1 g2 is mations and the generators of the Lie algebra, which satisfy a complex-valued antisymmetric function of the group ele- the following communication rules: ments with |ε(g1, g2)| = 1. Such representations are called [ 1, 2]= , projective representations. For the case of simply connected T T iE groups like the rotation group SO(3) projective representa- [T1, T3]=−iT2, ( ) tions are obtained by replacing the group SO 3 by its univer- [T2, T3]=iT1, sal cover SU(2). However, in the case of the non-semisimple [Ta, E]=0. (25) Euclidean group E(2) the covering group is not enough, one has to substitute this group by a larger group: the univer- Of particular interest are the generators on this group: c     sal central extension E (2), which includes in addition to ∂ ∂ ∂ ∂ 1 1 2 1 the group elements of E(2) the group U(1) of phases fac- T =−i + ξ2 , T =−i − ξ1 , ∂ξ 2 ∂β ∂ξ 2 ∂β tors ε(g , g ) with | ε(g , g ) |= 1. In general a central  1  2 1 2 1 2 ∂ ∂ ∂ extension Gc of a group G with elements (g, ς) ∈ Gc and T3 =−i ξ − ξ , E =−i . 1 ∂ξ 2 ∂ξ ∂β (26) g ∈ G,ς ∈ U(1) satisfies the group law [20] 2 1 The group Laplacian (Casimir operator) is determined by (g,ς) = (g1,ς1) ∗ (g2,ς2)  = ( 1)2 + ( 2)2 + 3 . = ((g1 ∗ g2,ς1ς2 exp[iξ(g1, g2)], (21) T T 2T E (27) where ξ(g1, g2) is the two-cocycle satisfying the relation Accordingly, the operator E is the center of the group. To 3 ξ(g1, g2) + ξ(g1 ∗ g2, g3) = ξ(g1, g2 ∗ g3) + ξ(g2, g3), derive a canonical basis we use the eigenfunction of E, T c ξ(e, g) = ξ(g, e) = 0. and the Laplace operator  of E (2). Using polar coor- The two-dimensional Euclidian group E(2) = T (2) ⊗ dinates ξ1 = ξ cos φ,ξ2 = ξ sin φ and hnm (ξ,β,φ) = SO(2) with E(2) ={(α, a) | α ∈ R,(mod 2π),a = exp(iβ)(exp(imφ)gnm (ξ) we find with the Laplacian (27) (a1, a2)T ∈ R2} is a semi-direct product of translations the following equation for gnm (ξ) :    and rotations of the two-dimensional Euclidian plane. Since ∂ ∂ 1 1 2 2 2 − ξ + m +  ξ − 2m g  (ξ) the invariant subgroup SO(2) is abelian, this group is not ξ ∂ξ ∂ξ ξ 2 nm semisimple and not compact and the unitary representations = nm gnm (ξ). (28) are infinite dimensional. The representations of the group E(2) constructed on the space of functions are well known. With the solutions of (28) we find for hnm (ξ,β,φ) The action of the group E(2) on a vector Z = (ξ ,ξ ) is |m|/2 1 1 h  (ξ,β,φ) = N  |  | exp(iβ)(exp(imφ) nm nm   given by 2 |  | ξ | | | | (α, )(ξ ,ξ ) = (ξ α−ξ α + 1, × exp − ξ m L m (|  | ξ 2), a 1 2 1 cos 2 sin a 2 n 2 ξ1 sin α+ξ2 cos α + a ). (22) (29)

 ! 1 The most general (projective) representations of E(2) can- = ( n ) 2 = ( + 1 + 1 ( + with Nnm π (|m|+n)! , nm 4 n 2 2 m not be obtained from its universal covering group but by the |m|), with n = 0, 1, 2,...,m = 0, ±1, ±2,..., = central extended group Ec(2). This group has been studied |m| , ±1, ±2,...Here Ln (x) are the associated Legendre poly- previously as e.g. in [20–22]. The central extension embod- nomials. The solutions hnm form an ortho-normalized set ies in addition a U(1) subgroup characterized by a complex and have a form analogous to the solutions of the 2D ζ = ( ω) c( ) (α, ,ω) parameter exp i . E 2 consists of elements a Schrödinger equation in the symmetric gauge for (α, ) ∈ ( ), ω ∈ c( ) with a E 2 R. The action of the group E 2 in a constant magnetic field [23]. The group Ec(2) is isomor- on a vector Z = (ξ ,ξ ,β)is described by [20–22] 1 2 phic to the quantum harmonic oscillator group studied e.g. 1 in [24]. (α, a, ω)(ξ1,ξ2,β) = ξ1 cos α−ξ2 sin α + a , The special eigenvalue  = 0 plays a particular role. The ξ α+ξ α + 2, solution of (28)for = 0 is given by 1 sin 2 cos a  1 ρ 1 β + ω + m(α, a,ξ1,ξ2) , (23) h = √ J|m|(ρξ) exp(imφ), (30) 2 m00 2π 123 779 Page 6 of 15 Eur. Phys. J. C (2019) 79 :779 and it agrees with the results for the group E(2). J|m|(x) are the TB description, different from the two-component for- the Bessel functions. The eigensolutions fall into two classes: malism, the basic objects on the group SU(2) (which here is 2 the continuous series with ρ = 0 and the discrete series with interpreted as the isospin group) are functions (x, z1, z2) ρ2 = 0 and depending both on the coordinate x of the spacetime man- ifold and the complex coordinates z = z (θ,ϕ,ψ),z = 1 1 1 2 h0 (φ) = √ exp(imφ). (31) z (θ,ϕ,ψ). General functions on the group manifold can be 0m0 π 2 2 decomposed into the form As described later the generator T3 corresponds to the hyper- 2 j ( , , ) = φ ( ) j ( , ). charge generator in the SM. It is convenient to introduce for x z1 z2 j3 x f j z1 z2 (37) ± 1 3 T and T the generators T = √ (T1 ± iT2). The action j =0 1 2 2 3 + of the operator T on the eigenfunctions gnm increases the =±1 The standard description for j3 2 by two-component hypercharge from m to m + 1, but it simultaneously gen- T + functions (φ1(x), φ2(x)) is obtained from (37) by the pro- erates states with different family numbers n: T gnm = n − jection into the configuration space by using the relation i=0 Ai gi,m+1, . Correspondingly the generator T reduce − φ1(x) the hypercharge: T g  = n B g , − , . A and B are (x, z , z ) = (z , z ) . nm i=0 i i m 1 i i 1 2 1 2 φ ( ) (38) coefficients, respectively. The action of the operator product 2 x (T−T+ + T+T−) on the eigenfunctions is given by For finite-dimensional representations of the group SU(2) the above presented description by using coordinates on (T−T+ + T+T−)g  (ξ,φ,β)= q g  (ξ,φ,β), (32) nm B nm the group manifold z1 and z2 is equivalent to the multi- 1 with qB(n, m,)= 4[n + (1+|m |)]. component description in nonabelian field theory. How- 2 c( ) The generators J1, J2, J3 on the group SU(2) are well ever, for a noncompact group as the group E 2 a multi-  = ( 2 + 2 + 2) component representation is not favorable and the tangent known and, by using the Laplacian J1 J2 J3 on this group, all finite-dimensional representations can be bundle approach using coordinates of the tangent space is found (see e.g. [25,26]. The application of the operators J± = more convenient. −1/2 1 2 2 (J ±iJ ) and J3 on the eigenfunctions of the Laplacian  leads to 5 Lagrangians on the tangent bundle

j = 1[ ( + ) − ( ± ]1/2 j , 5.1 Lagrangian of Gauge fields on the group J± f. j j j 1 j3 j3 1 f j ±1 (33) 3 2 3 SU(2) ⊗ Ec(2) J f j = j3 f j , (34) 3 j3 j3 Based on the above described orbit decomposition and using = (− . − + ,..., ) where j j3 j3 1 j3 are the isospin quan- the group G = SU(2) ⊗ Ec(2) the covariant derivative (10) tum numbers, j3 is the projection on the third isospin axis. a with ωμ = 0 can be rewritten as Moreover, we find j j ∂ (J+J− + J−J+) f. = qW f. , (35) a 1 a j3 j3 Dμ = + ig Aμ(x, u)Ja + ig Bμ(x, u)Ta ∂xμ 1 2 2 =[ ( + ) − 2] = with qW j j 1 j3 . Using the parametrization z1 + ig3Cμ(x, u)E, (39) θ [ (ψ − ϕ)/ ], = θ [ (ψ + ϕ)] cos 2 exp i 2 z2 i sin 2 exp i we find for j = 1/2, j = 1/2 from the SU(2) Laplacian f j = z ( ) 3 j3 1 where Ja are the generators of the group SU 2 and Ta and E j c( ) a ( , ) a ( , ) and for j = 1/2, j3 =−1/2 : f = z2. For the isospin are the operators (26) on the group E 2 . Aμ x u , Bμ x u j3 √ = , = 1 = ( )2/ , = and Cμ(x, u) are the frame connection coefficients (gauge numbers j 1 j3 1wehave f1 z1 2 for j3 √ 0 one finds f 0 = z z , and j =−1 yields f 1 = (z )2/ 2. potentials). The gauge field strength tensors can be obtained 1 1 2 3 −1 2 [ , ] The eigenfunctions for the general case for the isospin j are from the commutators Dμ Dν . Let us first consider the c( ) given by [26] gauge fields for the group E 2 . The following relations a ( , ) ( , )   for the field strength tensors Bμν x u and Cμν x u of the 1/2 1 + − generators T and E can be derived: = j j3 j j3 . a f jj3 z1 z2 (36) ( j + j3)!( j − j3)!

In traditional quantum field theory the internal degrees of ± ∂ ± ∂ ± Bμν = Bν − Bμ freedom, such as isospin, hypercharge or color, are described ∂xμ ∂xν ± 3 3 ± by spacetime depending multi-component fields taking into ±g2i(Bμ Bν − Bμ Bν ), account the vertical structure given by the internal gauge ∂ ∂ 3 3 3 Bμν = Bν − Bμ, groups by the corresponding Lie-algebra representation. In ∂xμ ∂xν 123 Eur. Phys. J. C (2019) 79 :779 Page 7 of 15 779

∂ ∂ = − − ( + − − − +), with an arbitrary parameter k. The metric gab induces a fam- Cμν μ Cν ν Cμ g2i Bμ Bν Bμ Bν (40) ∂x ∂x ily of non-degenerate invariant quadratic forms of Lorentz- a ± −1/2 1 2 invariant vectors vμ: where Bμ = 2 (Bμ ± iBμ). c Recently the non-semisimple group E (2) was found to 1 1μ 2 2μ 3 3μ 3 4μ I = vμv + vμv + kvμv + 2vμv . (46) be relevant in 1 + 1 gravity theory [27], for the construction of string background in the Wess–Zumino–Witten (WZW) → 1 With a scaling of gab k gab and k 1 the invariant model, in 3D Chern–Simon theory [28–30], and in Yang– product is given by Mills theory [29]. Gauge theory for a group with generators μ [ , ]= c I = v3 v3 . (47) La and commutators La Lb ifab Lc requires a bilinear 0 μ form g which is symmetric, invariant with respect of gauge ab 3 c( ) transformations, and non-degenerate so that there exists an The generator T of the group E 2 corresponds to the inverse matrix. For semisimple groups the invariant bilinear hypercharge operator in the SM and the Lagrangian of the ( ) form is given by the Killing form and it is proportional to corresponding gauge field is described by the U 1 group: K = d c = δ the Kronecker symbol gab fac fbd ab. For the non- Ec 1 3 3μν c L =− Bμν B . (48) semisimple group E (2) the Killing form is degenerate and it g 4 is given by gK = δ . Nevertheless there exists another form ab a3 In the general case the invariant product can be diagonal- of an invariant product for this gauge group. Let us determine 3 1 3 4 ized by the transformation vμ = √ (cos αv μ − sin αv μ), the general conditions for a non-degenerate invariant product. 2 4 1 3 4 2 vμ = √ (sin αv μ + cos αv μ) with tan 2α = . Then the The Lagrangian of the gauge fields must be a quadratic 2 k a combination of the field strength tensor Bμν. Lorentz covari- quadratic form is given by ance restricts its form to a bμ I = gabvμv , 1 a bμν μ μ Lg =− gab Bμν B , (41) = v1 v1μ + v2 v2μ + v3 v3 − v4 v4 , 4 μ μ a3 μ a4 μ (49) with the invariant symmetric metric gab (gab = gba). Under 2 2 with a3 = k cos α + 2 cos α sin α, a4 =−k sin α + an infinitesimal gauge transformation the field strengths α α a = a k = α = π/ a a b c 2sin cos . With the choice 3 4 we get 0, 4 transform like δBμν =−f Bμνθ . The Lagrangian must bc and a3 = a4 = 1. be gauge-invariant, therefore the following condition has to Using (45) the gauge Lagrangian of the Ec(2) model is be fulfilled: given by a b dμν e gab Bμν f B θ = 0. (42) de 1 a bμν L Ec =− gab Bμν B Accordingly the metric gab must satisfy the following con- 4 dition: k 3 3μν 1 + −μν 3 μν =− Bμν B − (Bμν B + BμνC ). (50) c c 4 2 gac f + gbc f = 0. (43) bd ad The Lagrangian is invariant under the gauge transformations For the group Ec(2) with the commutation rule (25)forthe ± ± 3 ± 3 ± δB = ∂μθ ∓ g i(B θ − θ B ), generators La = Ta(a = 1, 2, 3) and L4 = E the nonzero μ 2 μ μ c 4 =−4 = , 1 = 3 3 coefficients fab are given by f12 f21 1 f23 δBμ = ∂μθ , − 1 = , 2 =− 2 =− f32 1 f13 f31 1. Accordingly one can derive a C + − − + δCμ = ∂μθ + g2i(θ Bμ − θ Bμ ), (51) non-degenerate symmetric invariant bilinear form gab = gba given by [27–29] where θ 3,θ±, θC are arbitrary spacetime depending func- ⎡ ⎤ 1000 tions. ⎢0100⎥ The Lagrangian (50) is not positive definite and leads to g0 = ⎢ ⎥ . (44) ab ⎣0001⎦ negative terms in the Hamiltonian and to the occurrence of 0010 particles with un-physical negative norm. This situation is analogical to the case of gauge field in covariant The most general invariant quadratic form is a linear com- gauge of the SM with the gauge group SU(2)⊗U(1), where 0 K = δ bination of the metric gab and the Killing form gab a3 due to the form of the Lorentz metric the time-like component given by ´a ⎡ ⎤ ⎡ ⎤ B0 must correspond to negative metric particles. Quantiza- 1000 100 0 tion requires one to choose a specific gauge by adding terms ⎢ ⎥ ⎢ ⎥ a 2 2 0100 010 0 like (∂μ Bμ) and (∂μCμ) to the Lagrangian for a covari- g = ⎢ ⎥ , gab = ⎢ ⎥ , (45) ab ⎣00k 1⎦ ⎣000 1⎦ ant gauge which breaks the gauge invariance and introduce 0010 001−k new un-physical fields. These so-called Fadeev–Popov 123 779 Page 8 of 15 Eur. Phys. J. C (2019) 79 :779

fields cancel the un-physical gauge field components with 1 μ − 2 1 μ 3 2 + (∂ Wμ ) + (∂ Aμ) negative norm. 2ξ 2ζ 3 3 + − − + A diagonal form of the Lagrangian (50) can be achieved −(∂μω )ω − ∂μω ω − (∂μω )ω by the transformations 3 + −μ − +μ +ig2(∂μω )(ω W − ω W ) + − + − − + 3μ Cμ = (C − C ) −ig [(∂μω )ω − (∂μω )ω ]B μ μ 2  3 + − Bμ = (Cμ + Cμ ) (52) + −μ − +μ 3 −ig2[(∂μω )W − (∂μω )W ]ω , (56) or the corresponding field strengths SU( ) W ± = + − with the field strength tensors of the 2 gauge field μ Cμν = (C − C ), −1/2 1 2 3 a a μν μν 2 (Aμ ±iAμ) and Aμ. ω and ω are a set of independent 3 + − Bμν = (Cμν + Cμν). (53) anticommuting variables of the ghosts.

The Lagrangian of gauge particles with a gauge fixing 5.2 Lagrangians of matter fields on the group term and the Fadeev–Popov ghosts is for k = 0 given by ( ) ⊗ c( )  SU 2 E 2 1 + −μν + +μν − −μν L Ec =− 2Bμν B + CμνC − CμνC 4 In the Lagrangian of the Higgs scalar particles L H = † μ (DμH ) (D H ) the electromagnetic field should not cou- 1 μ + 2 1 μ − 2 1 μ + 2 + (∂ Bμ ) + (∂ Bμ ) + (∂ Cμ ) 2ξ 2ξ 2ζ ple to the and should be diagonalized. Substitut- ing (39) into the Lagrangian of the Higgs new mix- 1 μ − 2 3 3 + (∂ Cμ ) − (∂μc )c ing terms appear and the diagonalization requires that the 2ζ fields A3 , B3 and Cμ have to be transformed to new fields −∂ + − − (∂ −) + μ μ μc c μc c expressed by the relations 4 4 4 + −μ − +μ −(∂μ ) + (∂μ )( − ) c c ig2 c c B c B 3 c + − − + +μ −μ Bμ = cos θW Aμ − sin θW (cos θD Zμ − sin θD Eμ), −ig2[(∂μc )c − (∂μc )c ](C + C )  3 c Aμ = sin θW Aμ + cos θW (cos θD Zμ − sin θD Eμ), + −μ − +μ 3 −ig2[(∂μc )B − (∂μc )B ]c , (54) 3 c Cμ = sin θD Zμ + cos θD Eμ, (57)

± √1 1 2 ± 3 4 θ = θ , = where c = (c ± ic ), c3, c4 and c , c , c are the anti- where W is the , g2 g sin W g1 2 θ = (( )2 + ( )2)1/2 = θ θ commuting ghost fields and ζ is the gauge parameter. The g cos W , g g1 g2 , e g cos W sin W and θ θ =  /[(  )2 −( m H )2] cancellation of the un-physical gauge particles with negative D is defined by tan 2 D gg3m H H g3 H 2 g ,  norm by the Fadeev–Popov ghost particles can be proven by m H and H are the IQNs of the Higgs particle. The covariant the BRST symmetry in a gauge-invariant form. Quantization derivative (39) can be rewritten as of non-semisimple gauge groups has been studied in [29,31– ∂ + − + Dμ = + i[Wμ Q − + Wμ Q + ) + Bμ Q − 33]. In [29] one-loop radiative corrections for the Yang–Mills ∂xμ W W B c( ) − c model with the E 2 gauge group were computed. It was +Bμ QB+ + ZμQZ + AμQ + EμQEc ], (58) shown that there is no two- and higher-loop re-normalization ± = −1/2( 1 ± 2) ± = −1/2( 1 ± 2 ) and the full quantum effective action is given by the one-loop with Wμ 2 Wμ iWμ and Bμ 2 Bμ iBμ . term with the divergent part that can be eliminated by a field The following operators are introduced: redefinition.   1 The field strength tensor for the group SU(2) is given by Q = J + T , 3 2 3 ∂ ∂   ± = ± − ± Wμν μ Wν ν Wμ 2 2 1 ∂x ∂x QZ = g cos θD cos θW J3 − sin θW T3 ± 3 3 ± 2 ±g1i(Wμ Bν − BμWν ), + g sin θ E, ∂ ∂ 3 D 3 = 3 − 3 − ( + − − − +). Bμν μ Bν ν Bμ g1i Wμ Wν Wμ Wν ± 1 ± ∂x ∂x QW ± = g1J , QB± = g2 T , (55)  2  2 2 1 QEc = g − cos θW J3 + sin θW T3 sin θD The total gauge Lagrangian on the TB is Lg = L Ec + 2 L SU(2) where the Lagrangian of the group SU(2) with the + g3 cos θDE. (59) inclusion of ghost fields is given by  Any scalar function (x, u) defined on the fiber bundle =−1 + −μν + 3 3μν + 1 (∂μ +)2  L SU(2) 2Wμν W Aμν A Wμ  = (φ ( )χ ( ) + 4 2ξ can be expanded into the form M M x M u 123 Eur. Phys. J. C (2019) 79 :779 Page 9 of 15 779

φ† ( )χ ∗ ( )) − , = , } M x M u depending on the coordinates of the spacetime 2 j 0 E R and with the IQN of the Higgs particle: χ ( ) = (ξ,φ,β) ={ , = , =−1 . } manifold x and the eigenfunctions M u hlm MH n H m H 1 j3 2 H . An assumption con- ( , ) ( ) ⊗ c( )  f jj3 z1 z2 of the Laplacian of the group SU 2 E 2 cerning the IQN of and the Higgs particles will be defined by (29) and (36). Besides we introduced the follow- discussed later. ing symbol for the IQNs: M = (n, m,, j, j3). The total The self-interaction term V (S) and LYuk do not arise Lagrangian can be presented by in the TB tree-level approximation but are included for phe- nomenological reasons in the same way as in the SM. The L = L + L + L + L . (60) l H g Yuk microscopic foundation of these phenomenological terms is

Here the Lagrangian of the leptons Ll is defined in the chiral an unsolved problem in the SM as well as in the approach representation as here presented.

L = il†(x, u)σ μ l s s ∂ + − × + i[W Q − + W Q + ∂ μ μ W μ W 6 Quantization on the tangent bundle x  + + + c ] l ( , ), ZμQZ AμQ EμQEc s x u (61) On the tangent bundle one-particle states of the Dirac  ( , ) = (ψ ( )χ ( ) + ψ† ( )χ ∗ ( )) field f x u M,s M,s x M u M,s x M u μ μ labeled by the three-momentum p are described by ={ , } σ = (σ 0,σi ), σ = with the helicity s L R and R L (σ 0, −σ i ) l ( , ) = (ψ− ( )χ ( ) + ψ+ and s x u M M,s x M u M,s (x)χ ∗ (u)) M . Note that gauge fields and fermions or scalars  ( , ) = 1 [ f f ( )χ f ( ) ( ) f x u aK u M p M u exp ipx can carry different IQNs. As shown below, the known SM f ± K 2E V gauge particles as the Z and W bosons and the Ec gauge M ∗ boson carry the IQN  = 0, but due to the existence of fam- + † f v f ( )χ f ( ) (− )], bK M p M u exp ipx (64) ilies of SM leptons they carry nonzero Ec-charges  = 0. The interaction of leptons with the B± bosons is forbidden because of a selection rule, as discussed below. where the index s characterizes the helicity s ={L, R} and ={ , , } f ( ) The Lagrangian of the SM Higgs particles H = K M p s . aK t is the annihilation operator for a par-  ( , ) =−/ , = ) † f ( ) H x u (with I3 1 2 m 1 in the unitary gauge ticle in the interaction representation and bK t the antipar- is given by ticle creation operator satisfying the anti-commutation rules. f f ( ) EM is the single particle energy and V the volume. u M p † μ 2 f L H = ∂μ ∂ H +|H | and v (p) are the plane wave solutions of the  H M 2 for particles and antiparticles, respectively and the eigen- g1 + μ− 2 + μ− c cμ 2 × Wμ W + g qB Bμ B + Eμ E | Q Ec | χ ( ) 2 2 functions M u are given in (29) and (36). The general  construction of states in the TB indicates that not only 2 μ +|Q Z | Zμ Z + V (H ), (62) leptons but also scalars and gauge bosons carry the IQN MA ={n, m,, j, j3}. This means that scalar fields and 1 gauge fields can be expanded in a form analogous to (64). with Q =− cos θ g+g  sin θ , Q c = g sin θ /2+ Z 2 D 3 H D E D The structure of the theory based on the TB geometry sug-  θ , =  ( + ) g3 H cos D qB 4 H n 1 . The interaction of the SM gests the identification of an as a state Higgs with the W bosons remains the same as in the SM. c ± with specific internal quantum numbers M and a specific The E and B gauge bosons are not decoupled from the mass analogous to of atoms where dis- SM particles; according to (62) there is a coupling of the c ± crete quantum states with different quantum numbers and new E and Bμ bosons to the SM . In (62)the energy levels exist. Therefore we do not fix the particle con- ( ) =−μ2 |  |2 SM Higgs self-interaction potential V H H tent from the beginning but allow for the existence of “exotic” +λ |  |4 H is included. particles which do not appear in the SM and have not been = We denote the left- handed family by EL observed so far. The potential observation of such particles ( ,μ ,τ ) = ( , eL L L and the right-handed family by ER eR depends on its parameters, such as mass and lifetime, but also μ ,τ ) R R . The SM LYuk term is given by on selection rules as discussed below. =− σ †   + . . Inserting the expansion (64) into the fermion Laplacian LYuk n1n2 n1n2 M MH N h c (63) EL ER (61) the Hamiltonian is easy to build with an interaction term σ with n1n2 as constant coupling coefficient and with the IQNs with gauge particles g. For the unperturbed fermion Hamil- ={ , =− , = = 1 , } ={ , = MEL n1 m 1 j j3 2 EL , NE R n2 m tonian we get 123 779 Page 10 of 15 Eur. Phys. J. C (2019) 79 :779

2 2 2 1 f 1 † f f f f † with q = Q , q = Q , q c = Q , Q = e( j + m), H = p[a a − b b ], (65) A A Z Z E Ec A 3 2 0 2 P P P P = θ ( 2 θ − 2 θ 1 )+ θ , = P Q Z g cos D cos W j3 sin W 2 m g3 sin D qB 1 2 4[n + (1+|m |)], q =[j( j + 1) − j ], Q c = ={ , , } = 2 W 3 E with P M p s and with the one-particle energy p (− 2 θ + 2 θ 1 ) θ + θ  | | g cos W j3 g sin W 2 m sin D g3 cos D and p . For a compact representation we introduce for the gauge  = ( 0, c) 0 = ( , , c) particles the notation g g g with g A Z E S S∗ S g∗ g c ± ± I = dμ(u)χ (u)χ (u)χ (u)χ (u). (72) and g = (W , B ) and for the gauge potentials Agμ = MS Mg MS MS Mg Mg + − ± ± ± + − ∗ A μ + A μ with A μ = a μ(x)χ (u), χ = (χ ) .For g g g g Mg Mg Mg The dependence of the quantized field operators (64)on f the interaction Hamiltonian HI one gets the eigenfunctions χM (u) and on the coordinates u of the tan- gent vectors is a specific trait of the approach, here presented,

0 0 0 0∗ c c H f = (cg V g + cg †V g + ag V g based on the underlying geometric structure of the TB. The I K PRK K PRK K PRK PRK,g internal symmetries arise here from the inherent geometrical c c∗ ∗ symmetries of the TB in an analogous way to the symmetries + bg †V g )a f †a f + (cg U g + cg†U g K PRK P R K PRK K PRK in quantum mechanics originating from spacetime symme- + gc gc + gc† gc∗ ) f f †, tries in a given physical system. This differs in a principal aK UPRK bK U  bRbP (66) PRK way from standard QFT, therefore we call the theory here  presented a tangent bundle quantum field theory. √ with R ={N, r, s}, K ={Mg, k,λ}, K ={Mg, −k,λ} and Finally, we consider the SM Yukawa interaction LYuk f μ f f term given by (63). Inserting the expansion (64)into(63) u σs u μ(k)I δr,p−k g = P R MNMg , one gets expressions analog to the SM but including matrix VPRK (67) f f g elements 2 2Ep Er Ek V  ∗ μ Y = μχ H ( )χl ( )χl ( ). f f f IM M N d M u M u N u (73) v σs v μ(k)I δr,p+k H L R H L R g = P R MNMg . UPRK (68) f f g The TB eigenfunctions differ for different n and  but from 2 2Ep Er E V k (73) we see that the Yukawa interaction is nonzero only if not Here we introduced the matrix elements only leptons and but also the Higgs particles carry  c Y a nonzero E -charge H . The matrix element I is f g f ∗ f MH ML NR I = dμχ (u)χ (u)Q χ (u), (69) − +  + Mg M f N f Mg M f g N f nonzero if the relations for the hypercharges m EL m = c − +  +  = m E R 0 and for the E -charges EL  E R 0 with the integration measure dμ(u) = dμSU(2)dμEc , are fulfilled. The solution of these equations is not unique. 2 −1 2 −1 dμSU(2) = (16π ) sin θdθdψdϕ and dμEc = (4π ) Since interaction processes favor the lowest magnitude of ξdξdφdβ. The compact representation (66) includes all pos-  and m, we assume here the special solution m =  for sible interactions with gauge bosons g = A, Z, W ±, Ec and the above given two equations. For the left-handed lepton ±  =−, B . family we obtain EL 1 for the right-handed lepton  =−  = The interaction Hamiltonian of scalar particles in the uni- family E R 2 and for the Higgs family H 1. An tary gauge is described by important consequence is that with H = 0 analogous as lepton families also Higgs families should exist. H S =− (cS†cS + cS cS†) I P Q P Q PQKR 0 0 0 0 0 7 Lepton families, lepton universality and Higgs ×  [(cg †cg + cg cg †)M g g K R K R PQKR interaction beyond the Standard Model + ( gc† gc + gc gc†) gc ]. aK aR b  b MPQKR (70) K R 7.1 Lepton interaction and lepton universality Here we introduced the symbols P ={M , p}, P = S SM leptons are distinguished by the IQNs of isospin j and {M , −p}, Q ={M , q}, Q ={M , −q}, K ={M , k,λ},, S S S g j and m. Leptons consist of three fam- K ={M , −k,λ}, R ={M , r,σ}, R ={M , −r,σ} and 3 g g g ilies, electrons and are members of the the matrix elements first family, muons and muon neutrinos of the second and I S δλσ δ(p − q + k − r) taus and tau neutrinos of the third family. Different families g = 1 MS Mg , MPQKR qg (71) exhibit in the SM identical IQNs and properties in the elec- 8 S S g g V Ep Eq Er Ek troweak interaction with the exception of its masses. In the TB approach in addition to isospin I and hypercharge m the 123 Eur. Phys. J. C (2019) 79 :779 Page 11 of 15 779

Ec-charge  and the family quantum number n exist. Here (72). Let us first discuss the of the Z and the up to now unexplained fact in the SM that three lepton W ± bosons with the SM Higgs particle. As explained above families exist differentiated only by its mass find an explana- the Ec-charges  of the Z and the W bosons are necessarily tion by the additional family quantum number n for a nonzero zero (Z = 0,W ± = 0). However, as described in Sect. 6 c c E -charge . Besides, a larger number of families than three the Higgs particles carry a nonzero E -charge (here H = 1 could exist, but its possible observation depends on the mass is assumed), therefore the coefficients qZ in (71)showavery of these states with n 3, or possibly other physical effects. small deviation from their values in the SM proportional to 2 The interaction of fermions via gauge potentials is g3. described by (66) with analog expressions to the SM but An interesting feature of the presented approach refers to with inclusion of the matrix elements I f given in (69) the interaction of the extra gauge bosons Ec and B± with Mg M f N f depending on the eigenfunctions of the Laplacian on the the SM Higgs described by (70), (71) and (72). As seen the group. From (69), selection rules can be derived for fermion coupling of the Higgs to Ec and B± bosons is allowed. interactions. If the value of the integral I f is zero the Note that the existence of a new is a common Mg M f N f interaction is forbidden. These selection rules arise in a sim- feature of many extensions of the SM (for a review see [37]). ilar way to the selection rules in atomic systems if the tran- In particular models with an extra gauge group U´(1) are sition moment integral is vanishing, which constrains the studied in a large number of papers (see e.g. [37–40]). possible transitions of a system from one quantum state to The possible existence of the Ec and B± bosons leads to another. From these matrix elements we see that in the elec- a fundamental fifth interaction. The parameter space of Ec ± troweak interaction leptons (with Me = Ne) couple only to and B masses and the coupling coefficient g3 (or the mixing and Z-bosons with g = 0, jg = 0, mg = 0 and angle θD) are constrained by existing data from experiments, therefore with h (u) ≡ 1wefindI l = Q and and they could be found in a similar way to U´(1) extended Mg MA Ml Ml I l = Q . This means that a family universal elec- models (see e.g. [34–36,52–56]). Many of these data hint MZ Ml Ml Z troweak coupling of leptons with photons and Z-bosons is to the assumption that the coupling coefficient g3 and the regained in the approach here presented . For the coupling of mixing angle θD are small: g3  1,θD  1. This means the families of left-handed charged leptons and neutrinos to that the fifth mediated by the Ec- ± the charged W -bosons we substitute for Ml the IQN of the boson is much weaker than the SM weak interaction. lef-handed charged leptons and for Nl that of the neutrinos. ± The operator QW ± = g1J shifts the isospin component j3 in such a way that the matrix element (69) is again indepen- 8 Dark Matter candidates dent on the family number n. Lepton flavor universality is one of the distinctive features of the SM and experiments have Astrophysical and cosmological observations show that the set stringent limits on processes that violate this universality. largest part of matter in our is constituted by Although now every family is connected with a different IQN unknown non-luminous particles, called DM particles, that n and different eigenfunctions, the coupling of the photons have a very weak interaction with the visible sector of to the leptons remains independent on the family number n. the universe. Such particles do not exist in the SM, but The same behavior we can find for the interaction with the there have been many attempts of an extension of the Ec-boson with SM leptons. The matrix element (69) with SM with possible DM candidates such as weakly interact- g = Ec is nonzero only if the Ec-boson carries the IQN ing massive particles (WIMPs), sterile neutrinos, the - values Ec = 0, jEc = 0, m Ec = 0. est neutralinos in super-symmetric models or axions (see The mass of leptons and its large difference for electrons, e.g. [41,42]). Recently alternative phenomenological mod- muons and τ-leptons as well as the nonzero neutrino masses els have been developed, like the dark sector model (see cannot be explained as a tree-level effect. But the occurrence e.g. [36,47–51]) or the Higgs portal model [46,47,52,53, of a family quantum number n and different eigenfunctions in 55,56]. the approach here presented could open up a route towards One of the most notable features of the generalization of its physical understanding beyond the tree level in a non- the SM by the gauge group SU(2) ⊗ Ec(2) is the possibil- perturbative treatment. This problem is beyond the scope of ity that DM candidates lie within the new gauge sector. In the present paper. the present approach for the derivation of the corresponding Lagrangians of DM particles no additional phenomenologi- cal model assumptions are required, but only the particle con- 7.2 The interaction of Higgs particles with gauge bosons tent with the choice of appropriate IQNs for the DM is neces- sary. An obvious way for the assignment of the IQNs to left- The interaction of scalar particles with gauge bosons is and right-handed dark fermions and dark scalars can be made described by (70) with matrix elements given in (71) and by the choice of zero hypercharge (m = 0) and isospin ( j = 123 779 Page 12 of 15 Eur. Phys. J. C (2019) 79 :779

) c  = D =  D†σ μ 0 but nonzero E -charge 0. As a result one may expect L f i Ms s that, similar to the SM leptons, DM fermions and DM scalars Ms  = , , ∂ are grouped in families with the IQN n 1 2 3. In the c D × + ig  (sin θ Zμ + cos θ Eμ)  , Laplacian (61), (62) and (63) we substitute for the lepton and ∂xμ 3 D D D Ms l ( , ) → l ( , ) +  D( , ) Higgs wavefunctions s x u s x u s x u and (75) H (x, u) → H (x, u) + S(x, u). According to (29)the c eigenfunctions of the Laplacian on the group SU(2)⊗ E (2) with MD = (n, 0,D, 0, 0). As one can see, different types c with j = m = 0 take the form of DM fermions with nonzero E -charges D are predicted. For every DM fermion with given Ec-charge  aDM  D  fermion family with n = 1, 2,...could exist which couple D χ = h  (ξ, φ) = (iβ) M 00 π exp to the SM Z gauge potential with the coupling coefficient   g  sin θ and to the Ecgauge potential with the coupling |  | ξ 2 3 D D × − 0(|  | ξ 2). coefficient g  cos θ . Analogously to Eq. (69) one can exp Ln (74) 3 D D 2 derive corresponding selection rules for the interaction of dark fermions with gauge bosons. With these extensions the Laplacian (60) now is substituted The construction of a gauge and Lorentz-invariant mass by L → LSM +L D where LSM describe the SM particles and term for DM fermions in a renormalizable Lagrangian can = f + S + Yuk LD LD LD LD includes the Lagrangians of dark be done in a similar way to the SM using a modified Yukawa fermions, dark scalars and the dark Yukawa term, respec- interaction term and different IQNs for right- and left-handed tively. In the following we discuss these Lagrangians. DM fermions. Since the SM Higgs particles carry isospin D and hypercharge a DM Yukawa interaction term LYuk cannot be constructed from the SM Higgs but instead a scalar DM 8.1 Dark vector gauge bosons with vanishing hypercharge and isospin ( j = 0, m = 0) but nonzero Ec-charge ( = 0). Therefore the Yukawa interac- In the present approach new vector bosons Ec and B± arise S tion term for scalar DM can be expressed as naturally by the geometric TB symmetry described by the group Ec(2). These particles can be interpreted as DM vec- D =− σ D †   + . . LYuk n1n2 n n M MS N h c (76) tor gauge bosons. The Lagrangian of the DM gauge bosons 1 2 DL DR + − isgivenby(54) where the fields Cμ and Cμ are related with where the sum is over the DM fermion family members c , c c the Aμ Zμ and the Eμ gauge potentials by the relation (52) with identical , and σ D are constant coupling coeffi- c ± n1n2 and (57). The E and B gauge bosons are not decoupled cients. For the vacuum IQNs of the dark scalar we assume from the SM particles, according to (62) there is a coupling to the IQNs MS ={nS = 0, mS = 0, j3 = 0,S = 1}. c the SM Higgs, but also the interaction of the E boson with This suggests that the following be assigned for the IQN leptons with a very small g3 is allowed. values for the family of left-handed and right-handed dark Note that leptons interact also directly with the Higgs parti- ={ , = , = = , =−} fermions: MDL n1 m 0 j j3 0 DL 1 , cles due to the Yukawa interaction in (63) and therefore via ND ={n2, m = 0, j = 0,D =−2}. c ± R R (62) they indirectly couple to the E and Bμ bosons. From (40), (50), (54) and (57) we can see that the non- c 8.3 Dark scalars Abelian DM vector bosons with the gauge potentials Eμ and B± interact with each other but also with the SM gauge μ The Lagrangian of DM scalars with j = 0, m = 0but bosons A, Z and W ±. nonzero  = 0 and the family number n is given by Note that the hypothesis of self-interacting DM (in con- trast to collisionless cold DM) enables to resolve a number of D D† μ D D 2 L = ∂μ ∂  +| | conflicts between observations and predictions of collision- S  S  S  S less DM simulations [43,44] and has also been assumed as 1 + μ− × g24 n + B B light thermal DM relicts [45]. 2 μ 2  μ 2 c cμ 2 +Zμ Z (g3 sin θD) +Eμ E ((g3 cos θD) 8.2 Dark fermions + (D, ), V S H (77) We assume that DM fermions with vanishing hypercharge = , = ) c  = ) (D, ) and isospin ( j 0 m 0 but nonzero E -charge ( 0 where V S H is the nonlinear Higgs-type potential for could exist. The Lagrangian of the family of DM fermions is the DM scalar including a possible coupling of the SM Higgs given by particle to the DM scalar: 123 Eur. Phys. J. C (2019) 79 :779 Page 13 of 15 779   (D, ) 2 V S H 2 0 2 1 M c = 2( ) g sin θD + g cos θD =−μ2 |  |2 +λ |  |4 +λ |  |2|  |2 . E H 2 3 S S S S SH H S + (  θ )2(0 )2, (78) 2 g3 cos D S (86) 2 2 0 2 0 2 M ± = 2g [2( ) + ( ) ]. (87) As seen in (77) coupling of DM scalars to the DM gauge B 2 H S vector bosons Ec and B± is allowed. But with a very small ± coefficient g3 sin θD there exists also a coupling to the SM The mass of the W bosons is identical to its value in Z boson arising from the diagonalization of the Laplacian the SM. For the Z-boson we obtain from (85) a mass with 2 for the Higgs particle. The coupling of a DM scalar particle a small deviation from the√ SM proportional to (g3) for = ( c, , ±)  : δ = (0 ) 2/ to gauge bosons g E Z B is described by (70), (71) g3 1 MZ H 4g3 2g, which can be used for the χ D( ) and (72) and by the eigenfunctions M u as given in (74). determination of experimental bounds of the coupling coeffi- In order to generate the and DM fermion mass cient g3. As seen in (87) with H = 1, m H = 1, n H = 0, the the potential for the scalar DM should develop a nonzero B± boson gets a mass proportional to the SM coupling con- (D, ) + ( ) 0 → ) VEV. Taking the extremum of V S H V H a stant g2. Without the VEV of the dark scalar (with S 0 ≺  = 0 ≺  = 0 ± = 0  . 0 = nonzero VEV H H and S S can be one gets MB H g22 116 24 GeV (using H 180 calculated: GeV). According to (86) the mass of the Ec boson given by 0 2 2 MEc  2 g3 for g3  1 is proportional to g3. 4μ λS − 2μ λSH S (0 )2 = H S , Albeit on a different theoretical basis, the described H 4λ λ − λ2 S H SH predictions show some similar features that arise in var- 2 2 4μ λH − 2μ λSH ious scenarios for DM physics denoted as the dark sec- (0 )2 = S H . (79) S λ λ − λ2 4 S H SH tor (see e.g. [36,47–51]), the vector Higgs portal (see e.g. [46,47,52,53,55,56] and the Z-portal (see e.g. [54,55]). Choosing the unitary gauge and expanding the Higgs field The dark sector hypothesis assumes that DM interacts only and the DM scalar around their VEVs by  = 0 +  , H H h through a new U (1) force with a hypothetical “dark pho-  = 0 +  the mass squared matrix for the SM Higgs D S S s ton” as gauge boson but SM matter does not interact directly and for the DM scalars is given by   with DM particles but can interact indirectly via a kinetic λ (0 )2 λ 0 0 2 = 2 H H SH H S . mixing term in the Lagrangian [48]. In the Higgs portal M λ 0 0 λ (0 )2 (80) SH H S 2 S S model a DM massive vector boson associated with a hidden U’ (1) symmetry couples to the SM Higgs and in the Z- This matrix can be diagonalized by portal DM model a DM fermion interacts directly with the ´h = cos βh + sin βs, SM Z-boson. Reference [55,56] reports that the Higgs portal

´s =−sin βh + cos βs. (81) model is compatible with the available data. Besides in [55] also acceptable regions of parameters for the Z-portal model We have the mixing angle for the interaction of DM fermions with the SM Z-boson 0 0 were reported. From these results similar conclusions can be λSH  tan β = H S (82) λ (0 )2 − λ (0 )2 + drawn for the acceptable parameter region of the Lagrangians S S H H (75) and (77). = (λ (0 )2 − λ (0 )2)2 + (λ 0 0 )2 A comparison of the approach here presented with the with S S H H SH H S .The masses for the diagonalized mass eigenstates are Higgs portal and the Z-portal models shows some common properties but also clearly distinct features and principal dif- = λ (0 )2 + λ (0 )2 + , Mh´ S S H H ferences. Whereas the majority of DM models can be con- = λ (0 )2 + λ (0 )2 − . sidered as a minimal extension of the SM based on a phe- Ms´ S S H H (83) nomenological model assumption to understand the mecha- Spontaneous symmetry breaking of the SM Higgs field nism of annihilation, scattering and possible decays of SM and the DM scalar leads to the generation of masses for the ± particles the idea of the present approach is to describe fun- gauge bosons Z, W as well as for the dark vector bosons damental interactions of SM particles as well as DM par- c ± E and B which contain contributions from the Higgs VEV ticles in a uniform way without phenomenological model as well as from the DM scalar VEV, assumption within the geometrical structure of the TB with 2 0 2 2 ( ) ⊗ c( ) M ± = ( ) g /2(84)the same symmetry group SU 2 E 2 . As seen in the W H 1  2 Lagrangian (62) the SM Higgs particle interacts with the dark 2 = (0 )2 1 θ − θ vector bosons Ec and B±, and in the Lagrangian (77)aDM MZ 2 H g cos D g3 sin D 2 scalar and in (75) a DM fermion with the SM Z boson. The + (  θ )2(0 )2, c ± 2 g3 sin D S (85) extra DM vector bosons E and B are nonabelian gauge 123 779 Page 14 of 15 Eur. Phys. J. C (2019) 79 :779

c bosons and interact with the SM gauge bosons. Due to the families requires that leptons carry a nonzero E -charge l . existence of families the SM leptons carry an Ec-charge. A A selection rule from the Yukawa interaction indicates that c nonzero Yukawa interaction term hints that also the Higgs also the Higgs particle carries the E -charge H = 1 and particle carries a nonzero Ec-charge  and therefore a fam- therefore in the present approach a family of Higgs parti- ily of Higgs particles could exist. The coupling of the Higgs cles is predicted. In contrast, SM and Ec gauge bosons carry c ± c and DM scalars in (62) and (77) with the E and B arise in zero E -charge, g = 0. The derived selection rule reveals an intrinsic way by the symmetry group, while interactions that a family universal coupling of leptons persists for the of the Z-Boson with the DM Ec and B± boson or a scalar interaction with the SM gauge bosons. DM particle arise due to the diagonalization of the Higgs An important prediction of the theory presented is the Lagrangian by the transformations (57). possibility of identifying candidate stable or unstable hypo- Above we discussed only DM candidates with zero isospin thetical DM fermions and DM scalars with zero hypercharge and hypercharge, but there exists the possibility that DM par- and zero isospin but nonzero Ec-charge  = 0 which should ticles basically interact by the weak coupling to SM particles. necessarily be grouped in different families with different This means they are electrically neutral but carry hypercharge family numbers n = 1, 2,... The new non-Abelian vector = ( + 1 ) = c ± and isospin satisfying the relation Q A e j3 2 m 0. bosons E and B can be interpreted as DM vector gauge = 1 bosons. These hypothetical bosons are not decoupled from This includes right-handed sterile neutrinos with j3 2 and m =−1. the SM particles, but there is a weak coupling to the SM Higgs and to the SM Z-boson. Moreover, also the coupling of leptons with the Ec bosons with a small coupling coef- ficient g3 is allowed. DM vector bosons interact with each 9 Conclusions other but also with the SM gauge bosons. Spontaneous sym- metry breaking of the SM Higgs and the DM scalar predicts The present paper is based on the hypothesis that the tangent not only the masses of the Z and W ± bosons but also that bundle is the underlying geometrical structure for the descrip- of the dark gauge bosons Ec and B±. The precondition of tion of the fundamental physical interactions. The internal nonzero Ec-charges  = 0 of DM fermions and DM scalars (gauge) symmetries are not inserted as an extra theory con- indicates that analogous to lepton families also DM fermion stituent given externally a priori by phenomenological rea- and DM scalar families should exist. sons like in the SM but emerge from the inherent geometrical Finally, the approach presented is linked with the geometri- structure of the TB with symmetries described by the group zation program of physics based on the single hypothetical SO(3, 1). Projective irreducible representations of this group principle that the tangent bundle with the symmetry group can be constructed by using the little groups SU(2), Ec(2) SO(3, 1)  T (3, 1) is the fundamental geometrical structure and SU(1, 1). Using the covariant derivative given by the for an unified description of all fundamental physical inter- operators on the transformation group G = SU(2) ⊗ Ec(2), actions. On the one hand as briefly explained in Sect. 3 the and the corresponding connection coefficients (gauge poten- tangent bundle is the geometrical fundament for teleparal- tials) a generalized theory of the electroweak interaction lel gravity gauge theory based on translational transforma- is derived. Since the SM arises (without phenomenological tions T (3, 1) of tangent vectors along the fiber axis [11– assumptions) as a limit case the presented approach answers 15] which is fully equivalent to the Einstein gravity theory. the question why the gauge group of the electroweak inter- On the other hand here a generalized theory of electroweak action in the SM is G = SU(2) ⊗ U(1). interaction and dark matter is presented based on the little In the TB approach wave functions depend on the space- groups of SO(3, 1). Therefore gravity, electroweak interac- time coordinates x as well as on the coordinates of the tan- tion and DM are described by the same fundamental geomet- gent fibers u. The well-known SM Z and W ± gauge bosons rical structure of the TB. Note that with can be found again but in addition new extra gauge bosons the gauge group SU(3) is in this frame still missing. The Ec and B± are predicted which constitute a fifth fundamen- color group SU(3) of cannot be tal interaction. In addition to the SM quantum numbers of described as a geometrical symmetry in the TB in the same isospin and hypercharge, the Ec-charge  and the family way as the SU(2) ⊗ Ec(2) group leaving the scalar product quantum number n exist. The existence of the family IQN (3) invariant. However, the SU(3) symmetry could be hid- n in the TB approach sheds light on the mysterious appear- den in the fundamental of the tangent bundle geometry in a ance of lepton families in the SM requiring a distinct IQN surprising way arising as an emergent symmetry similar to for every family. 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