  We critically revisit the issue of power-law running in models with extra
dimensions. The analysis is carried out in the context of a higher-dimensional
extension of QED, with the extra dimensions compactified on a torus. It is
shown that a naive $\beta$ function, which simply counts the number of modes,
depends crucially on the way the thresholds of the Kaluza-Klein modes are
crossed. To solve these ambiguities we turn to the vacuum polarization, which,
due to its special unitarity properties, guarantees the physical decoupling of
the heavy modes. This latter quantity, calculated in the context of dimensional
regularization, is used for connecting the low energy gauge coupling with the
coupling of the $D$-dimensional effective field theory. We find that the
resulting relation contains only logarithms of the relevant scales, and no
power corrections. If, instead, hard cutoffs are used to regularize the theory,
one finds power corrections, which could be interpreted as an additional
matching between the effective higher-dimensional model and some unknown, more
complete theory. The possibility of estimating this matching is examined in the
context of a toy model. The general conclusion is that, in the absence of any
additional physical principle, the power corrections depend strongly on the
details of the underlying theory. Possible consequences of this analysis for
gauge coupling unification in theories with extra dimensions are briefly
discussed.
