# 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,[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 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.[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.

## Triviality and the renormalization group

The first evidence of possible triviality of quantum field theories was obtained by Landau, Abrikosov, and Khalatnikov[3][4][5] who obtained the following relation of the observable charge gobs with the “bare” charge g₀,

$g_{obs}=\frac{g_0}{1+\beta_2 g_0 \ln \Lambda/m}~,$

(1)

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,

$g_0=\frac{g_{obs}}{1-\beta_2 g_{obs} \ln \Lambda/m}~.$

(2)

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.

The actual behavior of the charge g(μ) as a function of the momentum scale μ is determined by the full Gell-Mann–Low equation

$\frac{dg}{d \ln \mu} =\beta(g)=\beta_2 g^2+\beta_3 g^3+\ldots ~,$

(3)

which gives Eqs.(1),(2) if it is integrated under conditions g(μ) =gobs for μ = m and g(μ) = g₀ for μ = Λ, when only the term with $\beta_2$ is retained in the right hand side.

The general behavior of $g(\mu)$ relies on the appearance of the function β(g). According to the classification by Bogoliubov and Shirkov,[6] there are three qualitatively different situations:

1. if $\beta(g)$ has a zero at the finite value g*, then growth of g is saturated, i.e. $g(\mu)\to g*$ for $\mu\to\infty$;
2. if $\beta(g)$ is non-alternating and behaves as $\beta(g) \propto g^\alpha$ with $\alpha\le 1$ for large $g$, then the growth of $g(\mu)$ continues to infinity;
3. 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 the full theory (beyond its perturbation context), as can be seen by reductio ad absurdum. Indeed, if gobs 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 gobs → 0.

Formula (1) is interpreted differently, however, in the theory of critical phenomena. In this case, Λ and g₀ have a direct physical meaning, related to the lattice spacing and the coefficient in the effective Landau Hamiltonian. The trivial theory with gobs=0 is obtained in the limit m → 0, which corresponds to the critical point. Such triviality has a physical meaning and corresponds to the absence of interaction between large-scale fluctuations of the order parameter. The fundamental question arises whether such triviality holds for arbitrary (and not only small) values of g₀. This question was investigated by Kenneth G. Wilson using the real-space renormalization group,[7] and strong evidence for the affirmative answer was obtained. Subsequent numerical investigations of lattice field theory confirmed Wilson’s conclusion.

However, it should be noted that “Wilson triviality” signifies only that the β-function is non-alternating and does not have non-trivial zeros: it excludes only the case (a) in the Bogoliubov and Shirkov classification. The “true” quantum triviality is a stronger property, corresponding to the case (c). While “Wilson triviality” is confirmed by several 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 a serious unresolved question. Partial evidence in favor of its positive solution has appeared;[8][9][10] implications for the Standard Model and the resulting Higgs Boson mass bounds have also been discussed.[11][12]

## References

1. ^ R. Fernandez, J. Froehlich, A. D. Sokal (1992). Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory. Springer. ISBN 0-387-54358-9.
2. ^ D. J. E. Callaway (1988). "Triviality Pursuit: Can Elementary Scalar Particles Exist?". Physics Reports 167 (5): 241–320. Bibcode:1988PhR...167..241C. doi:10.1016/0370-1573(88)90008-7.
3. ^ L. D. Landau, A. A. Abrikosov, and I. M. Khalatnikov (1954). Doklady Akademii Nauk SSSR 95: 497.
4. ^ L. D. Landau, A. A. Abrikosov, and I. M. Khalatnikov (1954). Doklady Akademii Nauk SSSR 95: 773.
5. ^ L. D. Landau, A. A. Abrikosov, and I. M. Khalatnikov (1954). Doklady Akademii Nauk SSSR 95: 1177.
6. ^ N. N. Bogoliubov, D. V. Shirkov (1980). Introduction to the Theory of Quantized Fields (3rd ed.). John Wiley & Sons. ISBN 978-0-471-04223-5.
7. ^ K. G. Wilson (1975). "The Renormalization Group: Critical phenomena and the Kondo problem". Reviews of Modern Physics 47: 4. Bibcode:1975RvMP...47..773W. doi:10.1103/RevModPhys.47.773.
8. ^ Callaway, D.; Petronzio, R. (1987). "Is the standard model Higgs mass predictable?". Nuclear Physics B 292: 497. Bibcode:1987NuPhB.292..497C. doi:10.1016/0550-3213(87)90657-2.
9. ^ I. M. Suslov (2010). "Asymptotic Behavior of the β Function in the φ4 Theory: A Scheme Without Complex Parameters". Journal of Experimental and Theoretical Physics 111 (3): 450. arXiv:1010.4317. Bibcode:2010JETP..111..450S. doi:10.1134/S1063776110090153.
10. ^ Frasca, Marco (2011). "Mapping theorem and Green functions in Yang-Mills theory". The many faces of QCD. Trieste: Proceedings of Science. p. 039. arXiv:1011.3643. Retrieved 2011-08-27.
11. ^ Lindner, M. (1986). "Implications of triviality for the standard model". Zeitschrift für Physik C 31: 295. Bibcode:1986ZPhyC..31..295L. doi:10.1007/BF01479540.
12. ^ Urs Heller, Markus Klomfass, Herbert Neuberger, and Pavlos Vranas, (1993). "Numerical analysis of the Higgs mass triviality bound", Nucl. Phys., B405: 555-573.