In a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of a classical theory. If the only resulting value of the renormalized charge is zero, the theory is said to be "trivial" or noninteracting. Thus, surprisingly, a classical theory that appears to describe interacting particles can, when realized as a quantum field theory, become a "trivial" theory of noninteracting free particles. This phenomenon is referred to as quantum triviality. Strong evidence supports the idea that a field theory involving only a scalar Higgs boson is trivial in four spacetime dimensions,[1] but the situation for realistic models including other particles in addition to the Higgs boson is not known in general. Nevertheless, because the Higgs boson plays a central role in the Standard Model of particle physics, the question of triviality in Higgs models is of great importance.
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 these Higgs theories, the interactions of the Higgs particle with itself are posited to generate the masses of the W and Z bosons, as well as lepton masses like those of the electron and muon. If realistic models of particle physics such as the Standard Model suffer from triviality issues, the idea of an elementary scalar Higgs particle may have to be modified or abandoned.
The situation becomes more complex in theories that involve other particles however. 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. A deeper understanding of the physical meaning and generalization of the renormalization process, which goes beyond the dilatation group of conventional renormalizable theories, came from condensed matter physics. Leo P. Kadanoff's paper in 1966 proposed the "block-spin" renormalization group.[3] The blocking idea is a way to define the components of the theory at large distances as aggregates of components at shorter distances.
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. He was awarded the Nobel prize for these decisive contributions in 1982.
In more technical terms, let us assume that we have a theory described by a certain function
Z
\{si\}
\{Jk\}
Now we consider a certain blocking transformation of the state variables
\{si\}\to\{\tildesi\}
\tildesi
si
Z
\tildesi
\{Jk\}\to\{\tildeJk\}
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 with the "bare" charge, where is the mass of the particle, and is the momentum cut-off. If is finite, then tends to zero in the limit of infinite cut-off .
In fact, the proper interpretation of Eq.1 consists in its inversion, so that (related to the length scale) is chosen to give a correct value of,
The growth of with invalidates Eqs. and in the region (since they were obtained for) and the existence of the "Landau pole" in Eq.2 has no physical meaning.
The actual behavior of the charge as a function of the momentum scale is determined by the full Gell-Mann–Low equationwhich gives Eqs., if it is integrated under conditions for and for, when only the term with
\beta2
The general behavior of
g(\mu)
The latter case corresponds to the quantum triviality in the full theory (beyond its perturbation context), as can be seen by reductio ad absurdum. Indeed, if is finite, the theory is internally inconsistent. The only way to avoid it, is to tend
\mu0
As a result, the question of whether the Standard Model of particle physics is nontrivial remains a serious unresolved question. Theoretical proofs of triviality of the pure scalar field theory exist, but the situation for the full standard model is unknown. The implied constraints on the standard model have been discussed.[9] [10] [11] [12] [13] [14]