Gauge structure of Yang-Mills theories with extra dimensions

An effective Lagrangian for Yang-Mills theories with $n$ extra dimensions is constructed. We start from a field theory governed by the extra-dimensional Poincar\'e group $ISO(1,3+n)$ and the extended gauge group $SU(N,M^{4+n})$, characterized by an energy scale $\Lambda$ and 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. Such theories are connected by a canonical transformation that hides $ISO(1,3+n)\otimes SU(N,M^{4+n})$ into $ISO(1,3)\otimes SU(N,M^{4})$, and endows the KK gauge fields with mass. Using a set of orthogonal functions $\{f^{(\underline{0})},f^{(\underline{m})}(\bar x)\}$, generated by the Casimir invariant $\bar{P}^2$ associated with the translations subgroup $T(n)\subset ISO(n)$, the degrees of freedom of $ISO(1,3+n)\otimes SU(N,M^{4+n})$ are expanded via a general Fourier series, whose coefficients are the degrees of freedom of $ISO(1,3)\otimes SU(N,M^{4})$. These functions, corresponding to the projection on the coordinates basis $\{|\bar{x} \big >\}$ of the discrete basis $\{|0\big >,|p^{(\underline{m})}\big >\}$ generated by $\bar {P}^2$, are central in defining the effective theory. Components along the base state $f^{(\underline{0})}=\big <\bar x|0\big>$, identified as the standard Yang-Mills fields, do not receive mass at the compactification scale; components along excited states $f^{(\underline{m})}=\big <\bar x|p^{(\underline{m})}\big>$, corresponding to KK excitations, receive mass at this scale. Associated with any direction $|p^{(\underline{m})}\neq0\big >$ there are a massive gauge field and a pseudo-Goldstone boson. Resemblances of this mass-generating mechanism with the Brout-Englert-Higgs mechanism are stressed.