Abstract
The presence of an extra dimension of size $R\ensuremath{\equiv}{M}_{c}^{\ensuremath{-}1}$ introduces corrections of order $(\ensuremath{\mu}{/M}_{c})\ensuremath{\alpha}$ to the gauge and Yukawa couplings and accelerates their running at scales $\ensuremath{\mu}$ larger than ${M}_{c}.$ This could result in a grand unification scale ${M}_{X}\ensuremath{\approx}{20M}_{c}.$ We study the corrections at the two-loop level. We find corrections of order $(\ensuremath{\mu}{/M}_{c}){\ensuremath{\alpha}}^{2}$ for the gauge couplings and of order $(\ensuremath{\mu}{/M}_{c}{)}^{2}{\ensuremath{\alpha}}^{2}$ for the Yukawa couplings. Therefore, in the Yukawa sector one- and two-loop contributions can be of the same order below ${M}_{X}.$ We show that in the usual scenarios the dominant gauge and Yukawa couplings are decreasing functions of the scale, in such a way that $(\ensuremath{\mu}{/M}_{c})\ensuremath{\alpha}$ becomes approximately constant and two-loop contributions introduce just a $30%$ correction which does not increase with the scale.
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