[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.
Indeed, the result gobs = const(g0) can be obtained from the functional integrals only for g0 ≫ 1, while its validity for g0 ≪ 1, based on Eq.
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 Monte Carlo results [12] seems to confirm the qualitative validity of the Landau–Pomeranchuk arguments, although a different interpretation is also possible.
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.
For comparison, the maximum energies accessible at the Large Hadron Collider are of order 1013 eV, while the Planck scale, at which quantum gravity becomes important and the relevance of quantum field theory itself may be questioned, is 1028 eV.
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}.
The possible macroscopic states of the system, at a large scale, are given by this set of fixed points.
[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.