Banks–Zaks fixed point explained

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

\beta(g) = - b _{0} g ^{3} + b _{1} g ^{5} + l{O}(g ^{7)}

where

b _{0} and b _{1} are positive constants. Then there exists a value g = g _{\ast} such that \beta(g _{\ast)} =0 :g ^{2} _{\ast} = (b _{0})/(b _{1}) .

If we can arrange

b _{0} to be smaller than b _{1} , then we have g ^{2} _{\ast} < 1 . It follows that when the theory flows to the IR it is a conformal, weakly coupled theory with coupling g _{\ast} .

For the case of a non-Abelian gauge theory with gauge group

SU(N _{c)} and Dirac fermions in the fundamental representation of the gauge group for the flavored particles we have

b _{0} = (1)/(16 \pi ^{2}) (1)/(3) ( 11 N _{c-2N} _{f)} and b _{1} =- (1)/(\left( ( 16 \pi ^{2)} ^{2}) (34)/(3) N ^{2} _{c} - (1)/(2) N _{f\left(2} (N ^{2} _{c} -1)/(N _{c}) + (20)/(3) N _{c} \right) \right)

where

N _{c} is the number of colors and N _{f} the number of flavors. Then N _{f} should lie just below \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 N _{f} (which guarantees asymptotic freedom), (11)/(2) N _{c>N} ^{f> (34 N ^{3} _{c})/(( 13 N ^{2-3 )} _{c})} where the lower bound comes from requiring b _{1>0} . This way b _{1} remains positive while - b _{0} is still negative (see first equation in article) and one can solve \beta (g) = 0 with real solutions for g . The coefficient b _{1} was first correctly computed by Caswell, 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, .

Notes and References

1. Book: Terning, John. Modern Supersymmetry: Dynamics and Duality. Oxford University Press. 2006. 0198567634. Oxford.
2. Belavin . A.A.. Migdal. A.A.. 5 March 1974. Calculation of anomalous dimensionalities in non-Abelian field gauge theories. JETP Lett.. 19. 181.
3. Caswell . William E. . Asymptotic Behavior of Non-Abelian Gauge Theories to Two-Loop Order . Physical Review Letters . American Physical Society (APS) . 33 . 4 . 22 July 1974 . 0031-9007 . 10.1103/physrevlett.33.244 . 244–246. 1974PhRvL..33..244C .
4. Banks . T. . Zaks . A. . On the phase structure of vector-like gauge theories with massless fermions . Nuclear Physics B . Elsevier BV . 196 . 2 . 1982 . 0550-3213 . 10.1016/0550-3213(82)90035-9 . 189–204. 1982NuPhB.196..189B .