Banks–Zaks fixed point

In quantum chromodynamics (and also N = 1 superquantum chromodynamics) with massless flavors, if the number of flavors, N_f, is sufficiently small (that is small enough to guarantee asymptotic freedom), 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, then the fixed point is called a Banks–Zaks fixed point.

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

$\beta(g) = -b_0 g^3 + b_1 g^5 \,$

where b₀ and b₁ are positive constants. Then, there exists a value $g=g_\ast$ such that $\beta(g_\ast) =0$:

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

If we can arrange b₀ to be smaller than b₁, then we have $g^2_\ast <1$. It follows that the theory in the IR is a conformal, weakly coupled theory with coupling $g_\ast$.

For the case of QCD the number of flavors, N_f, should lie just below $\tfrac{11}{2}N_c$, where N_c is the number of colors, in order for the Banks–Zaks fixed point to appear.

References

• T. Banks and A. Zaks, Nucl.Phys. B196, 189 (1982).