Gauge structure of Yang-Mills theories with extra dimensions

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 Poincare group ${\rm ISO}(1,3+n)$ and by the extended gauge group ${\rm SU}(N,{\cal M}^{4+n})$, which is characterized by an unknown energy scale $\Lambda$ 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 ${\rm ISO}(1,3)$ and ${\rm SU}(N,{\cal M}^{4})$ is constructed. The transition between such theories is carried out via a canonical transformation that allows us to hide the extended symmetries ${\rm ISO}(1,3+n)\otimes {\rm SU}(N,{\cal M}^{4+n})$ into the standard symmetries ${\rm ISO}(1,3)\otimes {\rm SU}(N,{\cal M}^{4})$, and thus endow the Kaluza-Klein gauge fields with mass. Using a set of orthogonal functions $\{f^{(\underline{0})},f^{(\underline{m})}(\bar x)\}$, which is generated by the Casimir invariant $\bar {P}^2$ associated with the translations subgroup $T(n)\subset {\rm ISO}(n)$, the degrees of freedom of ${\rm ISO}(1,3+n)\otimes {\rm SU}(N,{\cal M}^{4+n})$ are expanded via a general Fourier series, whose coefficients are the degrees of freedom of ${\rm ISO}(1,3)\otimes {\rm SU}(N,{\cal M}^{4})$. It is shown that these functions, which correspond to the projection on the coordinates basis $\{|\bar{x} \big >\}$ of the discrete basis $\{|0\big >,|p^{(\underline{m})}\big >\}$ generated by $\bar {P}^2$, play a central role in defining the effective theory. It is shown that those components along the base state $f^{(\underline{0})}=\big <\bar x|0\big>$ do not receive mass at the compactification scale, so they are identified with the standard Yang-Mills fields; but components along excited states $f^{(\underline{m})}=\big <\bar x|p^{(\underline{m})}\big>$ do receive mass at this scale, so they correspond to Kaluza-Klein excitations. In particular, it is shown that associated with any direction $|p^{(\underline{m})}\neq0\big >$ there are a massive gauge field and a pseudo-Goldstone boson. Some resemblances of this mass-generating mechanism with the Brout-Englert-Higgs mechanism are stressed.