  We perform a detailed analysis of one-loop corrections to the self-energy of
the (off-shell) gauge bosons in six-dimensional N=1 supersymmetric gauge
theories on orbifolds. After discussing the Abelian case in the standard
Feynman diagram approach, we extend the analysis to the non-Abelian case by
employing the method of an orbifold-compatible one-loop effective action for a
classical background gauge field. We find that bulk higher derivative and
brane-localised gauge kinetic terms are required to cancel one-loop divergences
of the gauge boson self energy. After their renormalisation we study the
momentum dependence of both the higher derivative coupling h(k^2) and the {\it
effective} gauge coupling g_eff(k^2). For momenta smaller than the
compactification scales, we obtain the 4D logarithmic running of g_eff(k^2),
with suppressed power-like corrections, while the higher derivative coupling is
constant. We present in detail the threshold corrections to the low energy
gauge coupling, due to the massive bulk modes. At momentum scales above the
compactification scales, the higher derivative operator becomes important and
leads to a power-like running of g_eff(k^2) with respect to the momentum scale.
The coefficient of this running is at all scales equal to the renormalised
coupling of the higher derivative operator which ensures the quantum
consistency of the model. We discuss the relation to the similar one-loop
correction in the heterotic string, to show that the higher derivative
operators are relevant in that case too, since the field theory limit of the
one-loop string correction does not commute with the infrared regularisation of
the (on-shell) string result.
