In a quantum field theory, one may calculate an effective or running coupling constant that defines the coupling of the theory measured at a given momentum scale. One example of such a coupling constant is the electric charge.
In approximate calculations in several quantum field theories, notably quantum electrodynamics and theories of the Higgs particle, the running coupling appears to become infinite at a finite momentum scale. This is sometimes called the Landau pole problem.
It is not known whether the appearance of these inconsistencies is an artifact of the approximation, or a real fundamental problem in the theory. However, the problem can be avoided if an ultraviolet or UV fixed point appears in the theory. A quantum field theory has a UV fixed point if its renormalization group flow approaches a fixed point in the ultraviolet (i.e. short length scale/large energy) limit.[1] This is related to zeroes of the beta-function appearing in the Callan–Symanzik equation.[2] The large length scale/small energy limit counterpart is the infrared fixed point.
^Wilson, Kenneth G.; Kogut, John B. (1974). "The renormalization group and the ε expansion". Physics Reports. 12 (2): 75–199. Bibcode:1974PhR....12...75W. doi:10.1016/0370-1573(74)90023-4.
^Zinn-Justin, Jean (2002). Quantum Field Theory and Critical Phenomena. Oxford University Press.
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