The general formula for the anomalous dimension can be found in Martin΄s review
article (hep-ph/9709356), on page 62 relation (6.5.4). In the case of $SU(5)$ and especially in the paper of Kobayashi, Kubo, Mondragon and Zoupanos (hep-ph/9707425), which I am studying, I can deduce the first part of the relations (38), (page 15), which include the gauge coupling g.

However, I am extremely confused with the other part, which refers to the couplings $g^u$, $g^d$, $g^f$, $g^l$ in the superpotential $W$, relation (37). Is there anybody who can help me?
The first two terms in the superpotential are: $$\frac{1}{2} \ g^{u} \ 10 \ 10 \ H +g^{d} \ 10 \ \bar5 \ \bar H,$$ where $10$, $\bar 5$, are the antisymmetric and the antifundamental representations of $SU(5)$ for the fermions respectively and $H$, $\bar H$,the Higgs quintets and antiquintets. According to the paper (hep-ph/9707425), the anomalous dimension for the $10$ is: $$\gamma_{10}=\frac{1}{16\pi^{2}} \ (\frac{-36}{5} g^{2}+ 3 (g^{u})^{2} + 2 (g^{d})^{2}).$$ The general formula for the anomalous dimension is:
$$ \gamma_{i}^{j}=\frac{1}{2} Y_{ipq}Y^{jpq} -2 \delta_{i}^{j} C(i).$$ How can we deduce the factors in front of the gauge and Yukawa couplings?

This post imported from StackExchange Physics at 2014-11-29 18:09 (UTC), posted by SE-user ioannis