<PERSON>, <PERSON> & <PERSON> running formula
I've been told that for any group of SM, the running of the corresponding coupling constant, $g$, is given by:
$$ \frac{dg}{d(\ln{Q})} = b·g^3/(16\pi^2) $$
Where $$ b = -\frac{11}{3}C_2(A) + \sum\Bigg[\frac{2}{3}T(R_f) + \frac{1}{3}T(R_s) \Bigg] $$ and
$$ C_2(A) = \begin{cases} N, {\rm for\ }SU(N)\ 0, {\rm for\ }U(1) \end{cases}, \qquad T(R_f) = \begin{cases}\frac{1}{2}, {\rm spinors\ in\ fundamental\ repr.\ of\ }SU(N)\ N, {\rm spinors\ in\ adjoint\ repr.\ of\ }SU(N)\ Y_f^2, {\rm for\ }U(1) \end{cases} $$
$Y_f$ is the hypercharge of the corresponding field. $T(R_s)$ takes the same values as $T(R_f)$ for each complex scalar in the corresponding representation.
I'm trying to obtain the correct $b$ terms for each group, achieving:
$$ b(SU(3)) = -(11/3)·3 + 6·(2/3)·(1/2) + 3·(1/3)·(1/2) + 3·(1/3)·1 + 3·2·(1/6) + (1/6)·Y_B^2 + (1/3)·(1/2) = -6.3333... 346
$$
In the RHS and reading from left to right we find the term corresponding to $C_2(A)$, followed by the term for QCD quark triplets, SU(2) lepton doublets, SU(2) right parts, the 3 real bosons of SU(2) ($W_{1, 2, 3}$) the term for $B$ gauge boson for U(1) with zero hypercharge ($Y_B = 0$) and finally the <PERSON> contribution.
For SU(2):
$$ b(SU(2)) = -(11/3)·2 + 3·(2/3)·(1/2) + 9·(2/3)·(1/2) + (1/3)·8·3 + (1/6)·Y_B^2 + (1/3)·3·3·[(2/3)^2 + (1/3)^2] + (1/3)·(1/2) = 7.5 $$
Again, from left to right, $C_2(A)$ part, 3 lepton doublets, 3 quark lepton doublets with 3 colours each one, gluon contribution to $T(R_s)$, $B$ term, and 3 families with 3 colours for right quarks, 3 families for right charged leptons and <PERSON> term.
Checking with <PERSON> and <PERSON>, section 14.3, I know this 2 values are incorrect. Actually, for SU(2) we should have a result less than zero. What am I doing wrong?
Find this formula here: https://en.wikipedia.org/wiki/Beta_function_(physics)#SU(N)_Non-Abelian_gauge_theory 474