Quantum triviality

In a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of a classical theory. If the only allowed 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, 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 that may be accessible to present experimental efforts such as at the LHC. 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 predictable. 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.

Triviality and the renormalization group

The first evidence of possible triviality of quantum field theories was obtained by Landau, Abrikosov, Khalatnikov who obtained the following relation of the observable charge $g_{obs}$ with the “bare” charge $g_{0}$

$g_{obs}={\frac {g_{0}}{1+\beta _{2}g_{0}\ln \Lambda /m}}\qquad \qquad \qquad (1)$

where $m$ is the mass of the particle, and $\Lambda$ is the momentum cut-off. If $g_{0}$ is finite, then $g_{obs}$ tends to zero in the limit of infinite cut-off $\Lambda$. In fact, the proper interpretation of Eq.1 consists in its inversion, so that $g_{0}$ (related to the length scale $\Lambda ^{-1}$ ) is chosen to give a correct value of $g_{obs}$:

$g_{0}={\frac {g_{obs}}{1-\beta _{2}g_{obs}\ln \Lambda /m}}\qquad \qquad \qquad (2)$

The growth of $g_{0}$ with $\Lambda$ invalidates Eqs. 1 & 2 in the region $g_{0}\approx 1$ (since they were obtained for $g_{0}\ll 1$) and existence of the “Landau pole" in Eq.2 has no physical sense. The actual behavior of the charge $g(\mu )$ as a function of the momentum scale $\mu$ is determined by the Gell-Mann–Low equation

${\frac {dg}{d\ln \mu }}=\beta (g)=\beta _{2}g^{2}+\beta _{3}g^{3}+\ldots \qquad \qquad \qquad (3)$

which gives Eqs.1,2 if it is integrated under conditions $g(\mu )=g_{obs}$ for $\mu =m$ and $g(\mu )=g_{0}$ for $\mu =\Lambda$, when only the term with $\beta _{2}$ is retained in the right hand side. The general behavior of $g(\mu )$ depends on the appearance of the function $\beta (g)$. According to classification by Bogoliubov and Shirkov, there are three qualitatively different situations:

(a) if $\beta (g)$ has a zero at the finite value $g*$, then growth of $g$ is saturated, i.e. $g(\mu )\to g*$ for $\mu \to \infty$;

(b) if $\beta (g)$ is non-alternating and behaves as $\beta (g)\propto g^{\alpha }$ with $\alpha \leq 1$ for large $g$, then the growth of $g(\mu )$ continues to infinity;

(c) if $\beta (g)\propto g^{\alpha }$ with $\alpha >1$ for large $g$, then $g(\mu )$ is divergent at finite value $\mu _{0}$ and the real Landau pole arises: the theory is internally inconsistent due to indeterminacy of $g(\mu )$ for $\mu >\mu _{0}$.

The latter case corresponds to the quantum triviality in full theory (beyond its perturbation context), as can be seen by a reductio ad absurdum. Indeed, if $g_{obs}$ is finite, the theory is internally inconsistent. The only way to avoid it, is to tend $\mu _{0}$ to infinity, which is possible only for $g_{obs}\to 0$.

Formula (1) is interpreted differently in the theory of critical phenomena. In this case, $\Lambda$ and $g_{0}$ have a direct physical sense, being related to the lattice spacing and the coefficient in the effective Landau Hamiltonian. The trivial theory with $g_{obs}=0$ is obtained in the limit $m\to 0$, which corresponds to the critical point. Such triviality has a physical sense and corresponds to absence of interaction between large-scale fluctuations of the order parameter. The fundamental question arises, if such triviality holds for arbitrary (and not only small) values of $g_{0}$? This question was investigated by Kenneth G. Wilson using the real-space renormalization group and strong evidence for the positive answer was obtained. Subsequent numerical investigations of the lattice field theory confirmed Wilson’s conclusion.

However, it should be noted that “Wilson triviality” signifies only that $\beta$-function is non-alternating and has not non-trivial zeros: it excludes only the case (a) in the Bogoliubov and Shirkov classification. The “true” quantum triviality is a more strong property, corresponding to the case (c). If “Wilson triviality” is confirmed by numerous investigations and can be considered as firmly established, the evidence of “true triviality” is scarce and allows a different interpretation. As a result, the question of whether the Standard Model of particle physics is nontrivial (and whether elementary scalar Higgs particles can exist) remains an important unresolved question. The evidence in favour of its positive solution has appeared recently.