Landau pole
[2][3] The fact that couplings depend on the momentum (or length) scale is the central idea behind the renormalization group.Landau poles appear in theories that are not asymptotically free, such as quantum electrodynamics (QED) or φ4 theory—a scalar field with a quartic interaction—such as may describe the Higgs boson.Numerical computations performed in this framework seem to confirm Landau's conclusion that in QED the renormalized charge completely vanishes for an infinite cutoff.1, may be related to other reasons; for g0 ≈ 1 this result is probably violated but coincidence of two constant values in the order of magnitude can be expected from the matching condition.The case (c) in the Bogoliubov and Shirkov classification corresponds to the quantum triviality in full theory (beyond its perturbation context), as can be seen by a reductio ad absurdum.For example, QED is usually not believed[citation needed] to be a complete theory on its own, because it does not describe other fundamental interactions, and contains a Landau pole.The Higgs boson in the Standard Model of particle physics is described by φ4 theory (see Quartic interaction).Assume that we have a theory described by a certain function Z of the state variables {si} and a set of coupling constants {Jk}.[4] Solution of the Landau pole problem requires the calculation of the Gell-Mann–Low function β(g) at arbitrary g and, in particular, its asymptotic behavior for g → ∞.Application of more advanced summation methods yielded the exponent α in the asymptotic behavior β(g) ∝ gα, a value close to unity.The hypothesis for the asymptotic behavior of β(g) ∝ g was recently presented analytically for φ4 theory and QED.