In quantum chromodynamics (and also N = 1 superquantum chromodynamics) with massless flavors, if the number of flavors, Nf, 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 \( b_{0} \) and \( b_{1} \) 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 \( b_{0} \)to be smaller than \( b_{1} \), 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 \( SU(N_{c}) \) 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 \( N_c \) is the number of colors and \( N_{f} \) the number of flavors. Then \( N_{f} \) 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 \( N_{f} ( \)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 \( b_{1} \)remains positive while \( -b_{0} \) is still negative (see first equation in article) and one can solve \( {\displaystyle \beta (g)=0} \) with real solutions for g. The coefficient \( b_{1} \) 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

Terning, John (2006). Modern Supersymmetry: Dynamics and Duality. Oxford: Oxford University Press. ISBN 0198567634.

Belavin, A.A.; Migdal, A.A. (5 March 1974). "Calculation of anomalous dimensionalities in non-Abelian field gauge theories". JETP Lett. 19: 181.

Caswell, William E. (22 July 1974). "Asymptotic Behavior of Non-Abelian Gauge Theories to Two-Loop Order". Physical Review Letters. American Physical Society (APS). 33 (4): 244–246. doi:10.1103/physrevlett.33.244. ISSN 0031-9007.

Banks, T.; Zaks, A. (1982). "On the phase structure of vector-like gauge theories with massless fermions". Nuclear Physics B. Elsevier BV. 196 (2): 189–204. doi:10.1016/0550-3213(82)90035-9. ISSN 0550-3213.

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

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