Strong evidence supports the idea that a field theory involving only a scalar Higgs boson is trivial in four spacetime dimensions,[1][2] but the situation for realistic models including other particles in addition to the Higgs boson is not known in general.
This Higgs triviality is similar to the Landau pole problem in quantum electrodynamics, where this quantum theory may be inconsistent at very high momentum scales unless the renormalized charge is set to zero, i.e., unless the field theory has no interactions.
The Landau pole question is generally considered to be of minor academic interest for quantum electrodynamics because of the inaccessibly large momentum scale at which the inconsistency appears.
This is not however the case in theories that involve the elementary scalar Higgs boson, as the momentum scale at which a "trivial" theory exhibits inconsistencies may be accessible to present experimental efforts such as at the Large Hadron Collider (LHC) at CERN.
In fact, the addition of other particles can turn a trivial theory into a nontrivial one, at the cost of introducing constraints.
Depending on the details of the theory, the Higgs mass can be bounded or even calculable.
[2] These quantum triviality constraints are in sharp contrast to the picture one derives at the classical level, where the Higgs mass is a free parameter.
Quantum triviality can also lead to a calculable Higgs mass in asymptotic safety scenarios.
[2] Modern considerations of triviality are usually formulated in terms of the real-space renormalization group, largely developed by Kenneth Wilson and others.
Investigations of triviality are usually performed in the context of lattice gauge theory.
Leo P. Kadanoff's paper in 1966 proposed the "block-spin" renormalization group.
This approach covered the conceptual point and was given full computational substance[4] in Wilson's extensive important contributions.
The power of Wilson's ideas was demonstrated by a constructive iterative renormalization solution of a long-standing problem, the Kondo problem, in 1974, as well as the preceding seminal developments of his new method in the theory of second-order phase transitions and critical phenomena in 1971[citation needed].
The possible macroscopic states of the system, at a large scale, are given by this set of fixed points.
[2] The first evidence of possible triviality of quantum field theories was obtained by Landau, Abrikosov, and Khalatnikov[5][6][7] by finding the following relation of the observable charge gobs with the "bare" charge g0, where m is the mass of the particle, and Λ is the momentum cut-off.
If g0 is finite, then gobs tends to zero in the limit of infinite cut-off Λ.
In fact, the proper interpretation of Eq.1 consists in its inversion, so that g0 (related to the length scale 1/Λ) is chosen to give a correct value of gobs, The growth of g0 with Λ invalidates Eqs.
(1) and (2) in the region g0 ≈ 1 (since they were obtained for g0 ≪ 1) and the existence of the "Landau pole" in Eq.2 has no physical meaning.
According to the classification by Bogoliubov and Shirkov,[8] there are three qualitatively different situations: The latter case corresponds to the quantum triviality in the full theory (beyond its perturbation context), as can be seen by reductio ad absurdum.
Theoretical proofs of triviality of the pure scalar field theory exist, but the situation for the full standard model is unknown.