Conformal fixed point in certain Yang–Mills theories
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. 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 and by Caswell, and later used by Banks and Zaks 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
 formula_1
where formula_2 and formula_3 are positive constants. Then there exists a value formula_4 such that formula_5:
 formula_6
If we can arrange formula_2 to be smaller than formula_3, then we have formula_9. It follows that when the theory flows to the IR it is a conformal, weakly coupled theory with coupling formula_10.
For the case of a non-Abelian gauge theory with gauge group formula_11 and Dirac fermions in the fundamental representation of the gauge group for the flavored particles we have
formula_12
where formula_13 is the number of colors and formula_14 the number of flavors. Then formula_14 should lie just below formula_16 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 formula_14 (which guarantees asymptotic freedom), 
formula_18 
where the lower bound comes from requiring formula_19. This way formula_3 remains positive while formula_21 is still negative (see first equation in article) and one can solve formula_22 with real solutions for formula_23. The coefficient formula_3 was first correctly computed by Caswell, while the earlier paper by Belavin and Migdal has a wrong answer.
References.
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