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**Higher-dimensional routes to the Standard Model bosons**

Joao Baptista

*[Submitted on 6 May 2021]*

In the old spirit of Kaluza-Klein, we consider a spacetime of the form P = M_{4}×K, where K is the Lie group SU(3) equipped with a left-invariant metric that is not fully right-invariant. This metric has a U(1)×SU(3) isometry group, corresponding to the massless gauge bosons, and depends on a parameter ϕ with values in a subspace of su(3) isomorphic to C^{2}. It is shown that the classical Einstein-Hilbert Lagrangian density R_{P}−2Λ on the higher-dimensional manifold P, after integration over K, encodes not only the Yang-Mills terms of the Standard Model over M_{4}, as in the usual Kaluza-Klein calculation, but also a kinetic term |d^{A}ϕ|^{2}identical to the covariant derivative of the Higgs field.

For Λ in an appropriate range, it also encodes a potential V(|ϕ|^{2}) having absolute minima with |ϕ_{0}|^{2}≠ 0, thereby inducing mass terms for the remaining gauge bosons. The classical masses of the resulting Higgs-like and gauge bosons are explicitly calculated as functions of the vacuum value |ϕ_{0}|^{2}in the simplest version of the model. In more general versions, the values of the strong and electroweak gauge coupling constants are given as functions of the parameters of the left-invariant metric on K.

https://arxiv.org/abs/2105.02901

**Higher-dimensional routes to the Standard Model fermions**

Joao Baptista

*[Submitted on 6 May 2021]*

In the old spirit of Kaluza-Klein, we consider a spacetime of the form P = M_{4}×K, where K is the Lie group SU(3) equipped with a left-invariant metric that is not fully right-invariant. We observe that a complete generation of fermionic fields can be encoded in the 64 components of a single spinor over the 12-dimensional spacetime. The behaviour of the spinorial function along the internal space K can be chosen so that, after pairing and fibre-integration over K, the resulting Dirac kinetic terms in four dimensions couple to the u(1)⊕su(2)⊕su(3) gauge fields in the exact chiral representations present in the Standard Model. Although we describe the action of the internal Dirac operator on the 12-dimensional spinor, the full calculation of the fermionic mass terms produced by the model is longer and is not carried out here. We calculate instead the action of the internal Laplace operator on the spinor components.