Symmetry in the extra-dimensional Yang-Mills theory and its Kaluza-Klein effective description

Abstract

An effective Lagrangian for Yang-Mills theories with an arbitrary number of extra dimensions is constructed. We start from a field theory governed by the extra-dimensional Poincaré group ISO(1, 3+n) and by the extended gauge group SU (N, M 4+n ), which is characterized by an unknown energy scale Λ and is assumed to be valid at energies far below this scale. Assuming that the size of the extra dimensions is much larger than the distance scale at which this theory is valid, an effective theory with symmetry groups ISO(1, 3) and SU (N, M 4 ) is constructed. The transition between such theories is carried out via a canonical transformation that allows us to hide the extended symmetries {ISO(1, 3 + n), SU (N, M 4+n )} into the standard symmetries {ISO(1, 3), SU (N, M 4 )}, and thus endow the Kaluza-Klein gauge fields with mass. Using a set of orthogonal functions {f (0) , f (m) (x)}, which is generated by the Casimir invariant P 2 associated with the translations subgroup T (n) ⊂ ISO(n), the degrees of freedom of {ISO(1, 3 + n), SU (N, M 4+n )} are expanded via a general Fourier series, whose coefficients are the degrees of freedom of {ISO(1, 3), SU (N, M 4 )}. It is shown that these functions, which correspond to the projection on the coordinates basis {|x } of the discrete basis {|0 , |p (m) } generated by P 2 , play a central role in defining the effective theory. It is shown that those components along the base state f (0) = x|0 do not receive mass at the compactification scale, so they are identified with the standard Yang-Mills fields; but components along excited states f (m) = x|p (m) do receive mass at this scale, so they correspond to Kaluza-Klein excitations. In particular, it is shown that associated with any direction |p (m) = 0 there are a massive gauge field and a pseudo-Goldstone boson. Some resemblances of this mass-generating mechanism with the Englert-Higgs mechanism are stressed.

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

“…In the next sections, we implement the procedure outlined in Refs. [91,92] in order to obtain the {SO(1, 3), G SM (M 4 )}-invariant field theory, L d=4 eff , for the whole SM. This work complements the results given in these references on the extra-dimensional Yang-Mills sector with the corresponding Higgs and fermionic sectors.…”

“…According to Ref. [92], the effective Lagrangian with terms of canonical dimension d = 4 is derived from the straightforward extension of the SM as follows:…”

“…In Refs. [91,92], a thorough discussion of Yang-Mills theories in more than four dimensions and their subsequent compactification was presented. Here, we introduce notation and conventions, and present only the relevant results.…”

“…( 6), and integrating out N n using the orthogonality properties of f (m) E,O (see Appendix B of Ref. [92]), we obtain a four-dimensional Yang-Mills Lagrangian that is conveniently decomposed into the quantumchromodynamics (QCD) and electroweak (EW) sectors, that is,…”

“…The merger of both ideas is the very essence of the Englert-Higgs mechanism (EHM) [88][89][90]. In recent communications [91,92] by some of us, the symmetric structure of Yang-Mills theories with extra dimensions has been studied and its resemblance with the EHM stressed. In this work, we follow this approach closely to lay the foundations of a SM extension to extra flat dimensions, which we refer to by the acronym EDSM, for "extra-dimensional Standard Model".…”