The Standard Model in extra dimensions and its Kaluza-Klein effective Lagrangian

TL;DR

This paper develops an effective theory for the Standard Model extended with extra dimensions, describing how Kaluza-Klein modes acquire mass and their physical implications, including a detailed Lagrangian for calculations.

Contribution

It introduces a novel Fourier expansion approach to connect higher-dimensional theories with four-dimensional effective theories, detailing mass generation for KK modes.

Findings

KK excitations receive mass at the compactification scale

Ground state components correspond to Standard Model fields

Massive gauge fields and pseudo-Goldstone bosons emerge for excited states

Abstract

We construct an effective theory for the SM with extra dimensions. We start from a theory governed by the extra-dimensional groups ISO$(1,3+n)$ and $G_{\rm SM}({\cal M}^{4+n})=SU_C(3,{\cal M}^{4+n})\times SU_L(2,{\cal M}^{4+n})\times U_Y(1,{\cal M}^{4+n})$, characterized by an unknown energy scale $\Lambda$ and valid at energies far below this scale. Assuming that the extra dimensions are much larger than the distance scale of this theory, we construct an effective theory with symmetry groups ISO(1,3), $G_{\rm SM}({\cal M}^{4})$. The theories are connected by a canonical transformation that hides ISO$(1,3+n)$, $G_{\rm SM}({\cal M}^{4+n})$ into ISO(1,3), $G_{\rm SM}({\cal M}^{4})$; KK fields receive mass. Using a set of orthogonal functions $\{f^{(\underline{0})},f^{(\underline{m})}\}$, generated by the Casimir invariant $\bar{P}^2$ associated with the translations subgroup $T(n)\subset$ISO$(n)$, we expand the degrees of freedom of ISO$(1,3+n)$, $G_{\rm SM}({\cal M}^{4+n})$ in general Fourier series, whose coefficients are the degrees of freedom of ISO(1,3), $G({\cal M}^{4})$. These functions, which correspond to the projection on $\{|\bar{x}\big>\}$ of the discrete basis $\{|0\big>,|p^{(\underline{m})}\big>\}$ of $\bar{P}^2$, are central to define the effective theory. Components along the ground state $f^{(\underline{0})}=\big <\bar x|0\big>$ do not receive mass at the compactification scale, so they are the SM fields; components along excited states $f^{(\underline{m})}=\big<\bar x|p^{(\underline{m})}\big>$ get mass at this scale, so they are KK excitations. For any direction $|p^{(\underline{m})}\neq0\big>$ there are a massive gauge field and a pseudo-Goldstone boson. We stress resemblances of this mass-generating mechanism with the Englert-Higgs mechanism and discuss physical implications. We include a full catalog of Lagrangian terms that can be used to calculate Feynman rules.