Gauge-invariant approach to the beta function in Yang-Mills theories with universal extra dimensions

The radiative correction to the beta function is comprehensively studied at the one-loop level in the context of universal extra dimensions. Instead of using cutoffs to regularize one-loop divergences, the dimensional regularization scheme is used. In this approach, large momenta effects are removed from physical amplitudes by adjusting the parameters of the appropriate counterterms. The use of a SU(N)-covariant gauge-fixing procedure to quantize the theory is stressed. One-loop contributions of Kaluza-Klein (KK) excitations are characterized by discrete KK sums and continuous momenta sums, which can diverge. Two types of ultraviolet divergences are identified, one arising from poles of the gamma function and associated with short-distance effects in the usual four-dimensional spacetime manifold and the other emerging either from poles of the one-dimensional Epstein function or from the gamma function and corresponding to short-distance effects in the compact manifold. We address the cases of 5 and 4 + n dimensions (n >= 2) separately. In five dimensions, the one-dimensional Epstein function is convergent, so the usual counterterm renormalizes the vacuum-polarization function. For 4 + n dimensions, the one-dimensional Epstein function is divergent, so renormalization is implemented by interactions of canonical dimension higher than 4, already present in the effective theory. The polarization function is renormalized using both a mass-dependent scheme and a mass-independent scheme, with extra-dimensions effects decoupling in the former case but not in the latter. The beta function is calculated for an arbitrary number of extra dimensions. Our main result is that Yang-Mills theory remains perturbative at the one-loop level, which is in disagreement with the results obtained in the literature by using a cutoff regulator, which suggest that Yang-Mills theory in more than four dimensions ceases to be perturbative. We emphasize the advantages of a mass-dependent scheme in this type of theories, in which decoupling is manifest.