C-theorem: Difference between revisions

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*At fixed points of the RG flow, which are specified by a set of fixed-point couplings <math>g^*_i</math>, the function <math>C(g^*_i,\mu)=C_*</math> is a constant, independent of energy scale.

If such a so-called ''C''-function exists for a given quantum field theory, it specifies that the RG flow of the theory is irreversible.<ref>{{cite doi|10.1088/1742-5468/2010/10/P10004|noedit}}</ref>

[[Alexander Zamolodchikov]] proved in 1986 that two-dimensional quantum field theory always has such a ''C''-function. Moreover, at fixed points of the RG flow, which correspond to [[conformal field theory|conformal field theories]], Zamolodchikov's ''C''-function is equal to the [[central charge]] of the corresponding conformal field theory,<ref>[[Alexander Zamolodchikov|Zamolodchikov, A. B.]] (1986). [http://www.jetpletters.ac.ru/ps/1413/article_21504.pdf "Irreversibility" of the Flux of the Renormalization Group in a 2-D Field Theory], ''JETP Lett'' '''43''', pp 730-732.</ref> and roughly counts the degrees of freedom of the system.

Until recently, it had not been possible to prove an analog ''C''-theorem in higher-dimensional quantum field theory. However, in 2011, Zohar Komargodski and Adam Schwimmer proposed a proof for the physically more important four-dimensional case, which has gained acceptance.<ref>{{cite doi| 10.1038/nature.2011.9352|noedit}}</ref><ref name="komargodski">{{cite doi|10.1007/JHEP12(2011)099|noedit}}</ref> (Still, simultaneous monotonic and cyclic ([[limit-cycle]]) or even chaotic RG flows are compatible with such flow functions when multivalued in the couplings, as evinced in specific systems.<ref>{{cite doi|10.1103/PhysRevLett.108.131601|noedit}}</ref>)

as evinced in specific systems.<ref>{{cite doi|10.1103/PhysRevLett.108.131601|noedit}}</ref>

Also in 2011, Fortin, Grinstein and Stergiou discovered limit cycles and ergodic behavior in RG flows of unitary quantum field theories in <math>4-\epsilon</math> spacetime dimensions.<ref>{{cite doi|10.1016/j.physletb.2011.08.060|noedit}}</ref> (Conclusive examples in four dimensions have not yet been found.) These examples describe RG flows accessible in perturbation theory and thus do not have multi-valued ''C''-functions. Any possible ''C''-function is constant both in scale-invariant trajectories and at fixed points of the RG. These recurrent behaviors in the RG are associated with theories that are scale-invariant without being conformally invariant.

==See also==

*[[Conformal field theory]]