Eur. Phys. J. C (2019) 79:779 https://doi.org/10.1140/epjc/s10052-019-7260-z Regular Article - Theoretical Physics Extension of the standard model of electroweak interaction and Dark Matter 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 electroweak interaction In this paper we consider the general hypothesis that the is developed based on the underlying geometrical structure fundamental physical interactions 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 hypercharge, 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 bosons 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 fermions, 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 charge 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 gravity 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.