Banks–Zaks fixed point

More specifically, suppose that we find that the beta function of a theory up to two loops has the form

${\displaystyle \beta (g)=-b_0{g}^3+b_1{g}^5+\mathcal{O}(g^7)}$

where ${\displaystyle b_0}$ and ${\displaystyle b_1}$ are positive constants. Then there exists a value ${\displaystyle g=g_{\ast }}$ such that ${\displaystyle \beta (g_{\ast })=0}$:

${\displaystyle g_{\ast }^2=\frac{b_0}{b_1}.}$

If we can arrange ${\displaystyle b_0}$ to be smaller than ${\displaystyle b_1}$, then we have ${\displaystyle g_{\ast }^2<1}$. It follows that when the theory flows to the IR it is a conformal, weakly coupled theory with coupling ${\displaystyle g_{\ast }}$.

For the case of a non-Abelian gauge theory with gauge group ${\displaystyle SU(N_c)}$ and Dirac fermions in the fundamental representation of the gauge group for the flavored particles we have

${\displaystyle b_0=\frac{1}{16{\pi }^2}\frac{1}{3}(11N_c-2N_f)\text{ and }{b}_1=-\frac{1}{(16{\pi }^2{)}^2}\left(\frac{34}{3}{N}_c^2-\frac{1}{2}{N}_f\left(2\frac{N_c^2-1}{N_c}+\frac{20}{3}{N}_c\right)\right)}$

where ${\displaystyle N_c}$ is the number of colors and ${\displaystyle N_f}$ the number of flavors. Then ${\displaystyle N_f}$ should lie just below ${\displaystyle \frac{11}{2}{N}_c}$ in order for the Banks–Zaks fixed point to appear. Note that this fixed point only occurs if, in addition to the previous requirement on ${\displaystyle N_f}$ (which guarantees asymptotic freedom),

${\displaystyle \frac{11}{2}{N}_c>N_f>\frac{34N_c^3}{(13N_c^2-3)}}$

where the lower bound comes from requiring ${\displaystyle b_1>0}$. This way ${\displaystyle b_1}$ remains positive while ${\displaystyle -b_0}$ is still negative (see first equation in article) and one can solve ${\displaystyle \beta (g)=0}$ with real solutions for ${\displaystyle g}$. The coefficient ${\displaystyle b_1}$ was first correctly computed by Caswell, while the earlier paper by Belavin and Migdal has a wrong answer.