Standard Model with extra dimensions and its zeta function regularization
We start from a field theory governed by the extra-dimensional ISO(1,3+n) Poincar\’e group and by the extended SM gauge group, G(4+n). Then we construct an effective field theory whose symmetry groups are ISO(1,3) and G(4). The transition is carried out via two canonical transformations: a map that preserves, but it hides, the SO(1,3+n) symmetry; and a transformation, given by Fourier series, that explicitly breaks ISO(1,3+n) into ISO(1,3), but conserves and hides the gauge symmetry G(4+n), which manifests through nonstandard gauge transformations. From the 4-dimensional perspective, a particle that propagates in compact extra dimensions unfolds into a family of fields that reduces to the SM field if the size of the compact manifold is negligible. We include a full catalogue of Lagrangian terms that can be used to derive Feynman rules. The divergent character of the theory at one-loop is studied. A regularization scheme, based on the Epstein zeta function (EZF), is proposed to handle divergences coming from short distance effects in the compact manifold. Any physical amplitude can be written as an infinite series involving products of EZFs and powers of the compactification scale. The two possible scenarios m(m⎯⎯⎯)>m(0⎯⎯) and m(m⎯⎯⎯)<m(0⎯⎯), with m(0⎯⎯) and m(m⎯⎯⎯) the energy scale of the physical process and the compactification scale, are studied for arbitrary n. In the former scenario, amplitudes are given in terms of EZFs valued on the positive part of the real axis, with divergences arising from singularities of the EZFs. In the latter scenario, amplitudes involve EZFs valued on the negative part of the real axis, they are free of divergences, and they simplify considerably when they depend on the ratio m2(m⎯⎯⎯)/m2(0⎯⎯).