In quantum chromodynamics (and also N = 1 super quantum chromodynamics) with massless flavors, if the number of flavors, N_f, is sufficiently small (i.e. small enough to guarantee asymptotic freedom, depending on the number of colors), the theory can flow to an interacting conformal fixed point of the renormalization group.^[1] If the value of the coupling at that point is less than one (i.e. one can perform perturbation theory in weak coupling), then the fixed point is called a Banks–Zaks fixed point. The existence of the fixed point was first reported in 1974 by Belavin and Migdal^[2] and by Caswell,^[3] and later used by Banks and Zaks ^[4] in their analysis of the phase structure of vector-like gauge theories with massless fermions. The name Caswell–Banks–Zaks fixed point is also used.

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 ${\displaystyle {\tfrac {11}{2}}N_{c}}$ 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,^[3] while the earlier paper by Belavin and Migdal ^[2] has a wrong answer.

See also

• Beta function

References

• T. J. Hollowood, "Renormalization Group and Fixed Points in Quantum Field Theory", Springer, 2013, ISBN 978-3-642-36311-5.