Quantum trivialityFrom Wikipedia, the free encyclopedia

In a quantum field theory, charge screening can restrict the value of the observable "renormalized" charge of aclassical theory. If the only allowed value of the renormalized charge is zero, the theory is said to be "trivial" ornoninteracting. Thus, surprisingly, a classical theory that appears to describe interacting particles can, when realizedas a quantum field theory, become a "trivial" theory of noninteracting free particles. This phenomenon is referred toas quantum triviality. Strong evidence supports the idea that a field theory involving only a scalar Higgs boson istrivial in four spacetime dimensions,[1] but the situation for realistic models including other particles in addition to theHiggs boson is not known in general. Nevertheless, because the Higgs boson plays a central role in the StandardModel 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 theorymay be inconsistent at very high momentum scales unless the renormalized charge is set to zero, i.e., unless the fieldtheory has no interactions. The Landau pole question is generally considered to be of minor academic interest forquantum 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 atwhich a "trivial" theory exhibits inconsistencies may be accessible to present experimental efforts such as at theLHC. In these Higgs theories, the interactions of the Higgs particle with itself are posited to generate the masses ofthe W and Z bosons, as well as lepton masses like those of the electron and muon. If realistic models of particlephysics such as the Standard Model suffer from triviality issues, the idea of an elementary scalar Higgs particle mayhave to be modified or abandoned.

The situation becomes more complex in theories that involve other particles however. In fact, the addition of otherparticles can turn a trivial theory into a nontrivial one, at the cost of introducing constraints. Depending on the detailsof the theory, the Higgs mass can be bounded or even predictable.[2] These quantum triviality constraints are insharp contrast to the picture one derives at the classical level, where the Higgs mass is a free parameter.

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

(1)

where is the mass of the particle, and is the momentum cut-off. If is finite, then tends to zero in thelimit of infinite cut-off . In fact, the proper interpretation of Eq.1 consists in its inversion, so that (related to thelength scale ) is chosen to give a correct value of :

(2)

The growth of with invalidates Eqs. (1) and (2) in the region (since they were obtained for ) and existence of the “Landau pole" in Eq.2 has no physical sense. The actual behavior of the charge as a

function of the momentum scale is determined by the Gell-Mann–Low equation

(3)

which gives Eqs.(1),(2) if it is integrated under conditions for and for , when only the term with is retained in the right hand side. The general behavior of depends on

the appearance of the function . According to classification by Bogoliubov and Shirkov,[6] there are threequalitatively different situations:

a. if has a zero at the finite value , then growth of is saturated, i.e. for ;b. if is non-alternating and behaves as with for large , then the growth of

continues to infinity;c. if with for large , then is divergent at finite value and the real Landau pole

arises: the theory is internally inconsistent due to indeterminacy of for .

The latter case corresponds to the quantum triviality in full theory (beyond its perturbation context), as can be seenby a reductio ad absurdum. Indeed, if is finite, the theory is internally inconsistent. The only way to avoid it, isto tend to infinity, which is possible only for .

Formula (1) is interpreted differently in the theory of critical phenomena. In this case, and have a directphysical sense, being related to the lattice spacing and the coefficient in the effective Landau Hamiltonian. The trivialtheory with is obtained in the limit , which corresponds to the critical point. Such triviality has aphysical 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 ? This questionwas investigated by Kenneth G. Wilson using the real-space renormalization group[7] and strong evidence for thepositive answer was obtained. Subsequent numerical investigations of the lattice field theory confirmed Wilson’sconclusion.

However, it should be noted that “Wilson triviality” signifies only that -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 trivialityis a stronger property, corresponding to the case (c). If “Wilson triviality” is confirmed by numerous investigationsand can be considered as firmly established, the evidence of “true triviality” is scarce and allows a differentinterpretation. As a result, the question of whether the Standard Model of particle physics is nontrivial (and whetherelementary scalar Higgs particles can exist) remains an important unresolved question. The evidence in favor of itspositive solution has appeared recently[8][9][10][11] and the implications for the Standard Model and the resultingHiggs Boson mass bounds have been discussed in .[12]

See alsoHierarchy problem

References1. ^ 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 (http://adsabs.harvard.edu/abs/1988PhR...167..241C). doi:10.1016/0370-1573(88)90008-7 (http://dx.doi.org/10.1016%2F0370-1573%2888%2990008-7).

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 (http://adsabs.harvard.edu/abs/1975RvMP...47..773W).doi:10.1103/RevModPhys.47.773 (http://dx.doi.org/10.1103%2FRevModPhys.47.773).

8. ^ Callaway, D.; Petronzio, R. (1987). "Is the standard model Higgs mass predictable?". Nuclear Physics B 292:497. Bibcode:1987NuPhB.292..497C (http://adsabs.harvard.edu/abs/1987NuPhB.292..497C). doi:10.1016/0550-3213(87)90657-2 (http://dx.doi.org/10.1016%2F0550-3213%2887%2990657-2).

9. ^ I. M. Suslov (2008). "Renormalization Group Functions of the φ4 Theory in the Strong Coupling Limit:Analytical Results". Journal of Experimental and Theoretical Physics 107 (3): 413. arXiv:1010.4081(http://arxiv.org/abs/1010.4081). Bibcode:2008JETP..107..413S(http://adsabs.harvard.edu/abs/2008JETP..107..413S). doi:10.1134/S1063776108090094(http://dx.doi.org/10.1134%2FS1063776108090094).

10. ^ I. M. Suslov (2010). "Asymptotic Behavior of the β Function in the φ4 Theory: A Scheme Without ComplexParameters". Journal of Experimental and Theoretical Physics 111 (3): 450. arXiv:1010.4317(http://arxiv.org/abs/1010.4317). Bibcode:2010JETP..111..450S(http://adsabs.harvard.edu/abs/2010JETP..111..450S). doi:10.1134/S1063776110090153(http://dx.doi.org/10.1134%2FS1063776110090153).

11. ^ Frasca, Marco (2011). "Mapping theorem and Green functions in Yang-Mills theory"(http://pos.sissa.it/archive/conferences/117/039/FacesQCD_039.pdf) (PDF). The many faces of QCD(http://sites.google.com/site/facingqcd/). Trieste: Proceedings of Science. p. 039. arXiv:1011.3643(http://arxiv.org/abs/1011.3643). Retrieved 2011-08-27.

12. ^ Lindner, M. (1986). "Implications of triviality for the standard model". Zeitschrift für Physik C 31: 295.Bibcode:1986ZPhyC..31..295L (http://adsabs.harvard.edu/abs/1986ZPhyC..31..295L). doi:10.1007/BF01479540(http://dx.doi.org/10.1007%2FBF01479540).

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