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In a [[Physics:Quantum field theory|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,<ref>{{cite book | author1 = R. Fernandez |  author2 = J. Froehlich |  author3 = A. D. Sokal | year = 1992 | title = Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory | publisher = Springer | isbn = 0-387-54358-9 }}</ref><ref name="TrivPurs"/> 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 [[Physics:Quantum electrodynamics|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 predictable.<ref name="TrivPurs">
{{cite journal
 | author=D. J. E. Callaway
 | year=1988
 | title=Triviality Pursuit: Can Elementary Scalar Particles Exist?
 | journal=Physics Reports
 | volume=167
 | issue=5 | pages=241–320
 | doi=10.1016/0370-1573(88)90008-7
 }}</ref> 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.
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[[File:Quantum_book1_triviality_yellow.png|thumb|280px|Quantum triviality.]]
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==Triviality and the renormalization group==
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.<ref>L.P. Kadanoff (1966): "Scaling laws for Ising models near <math>T_c</math>", Physics (Long Island City, N.Y.) '''2''', 263.</ref> 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<ref>K.G. Wilson(1975): The renormalization group: critical phenomena and the Kondo problem, Rev. Mod. Phys. '''47''', 4, 773.</ref> 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 <math>Z</math> of the state variables <math>\{s_i\}</math> and a certain set of coupling constants <math>\{J_k\}</math>. This function may be a partition function, an action, a Hamiltonian, etc. It must contain the
whole description of the physics of the system.

Now we consider a certain blocking transformation of the state variables <math>\{s_i\}\to \{\tilde s_i\}</math>,
the number of <math>\tilde s_i</math> must be lower than the number of <math>s_i</math>. Now let us try to rewrite the <math>Z</math> function ''only'' in terms of the <math>\tilde s_i</math>. If this is achievable by a certain change in the parameters, <math>\{J_k\} \to \{\tilde J_k\}</math>, then the theory is said to be '''renormalizable'''. The most important information in the RG flow are its '''fixed points'''. The possible macroscopic states of the system, at a large scale, are given by this set of fixed points. If these fixed points correspond to a free field theory, the theory is said to be '''trivial'''. Numerous fixed points appear in the study of lattice Higgs theories, but the nature of the quantum field theories associated with these remains an open question.<ref name="TrivPurs"/>

==Historical background==

The first evidence of possible triviality of quantum field theories was obtained by Landau, Abrikosov, and Khalatnikov<ref>
{{cite journal
 |year=1954
 |title=On the Elimination of Infinities in Quantum Electrodynamics
 |journal=Doklady Akademii Nauk SSSR
 |volume=95 |pages=497
}}</ref><ref>
{{cite journal
 |author1=L. D. Landau |author2=A. A. Abrikosov |author3=I. M. Khalatnikov |name-list-style=amp |year=1954
 |title=Asymptotic Expressin for the Green's Function of the Electron in Quantum Electrodynamics
 |journal=Doklady Akademii Nauk SSSR
 |volume=95 |pages=773
}}</ref><ref>
{{cite journal
 |author1=L. D. Landau |author2=A. A. Abrikosov |author3=I. M. Khalatnikov |name-list-style=amp |year=1954
 |title=Asymptotic Expressin for the Green's Function of the Photon in Quantum Electrodynamics
 |journal=Doklady Akademii Nauk SSSR
 |volume=95 |pages=1177
}}</ref> by finding the following relation of the observable charge {{math|''g''<sub>obs</sub>}} with the "bare" charge {{math|''g''<sub>0</sub>}}, 
{{NumBlk|:|<math>g_\text{obs} = \frac{g_0}{1+\beta_2 g_0 \ln \Lambda/m}~,</math>|{{EquationRef|1}}}}
where {{mvar|m}} is the mass of the particle, and {{math|Λ}} is the momentum cut-off. If {{math|''g''<sub>0</sub>}} is finite, then {{math|''g''<sub>obs</sub>}} tends to zero in the limit of infinite cut-off {{math|Λ}}.

In fact, the proper interpretation of Eq.1 consists in its inversion, so that {{math|''g''<sub>0</sub>}} (related to the length scale {{math|1/Λ}}) is chosen to give a correct value of {{math|''g''<sub>obs</sub>}},
{{NumBlk|:|<math>g_0=\frac{g_\text{obs}}{1-\beta_2 g_\text{obs} \ln \Lambda/m}~.</math>|{{EquationRef|2}}}}

The growth of {{math|''g''<sub>0</sub>}} with {{math|Λ}} invalidates Eqs. ({{EquationNote|1}}) and ({{EquationNote|2}}) in the region {{math|''g''<sub>0</sub> ≈ 1}} (since they were obtained for {{math|''g''<sub>0</sub> ≪ 1}}) and the existence of the "Landau pole" in Eq.2 has no physical meaning.

The actual behavior of the charge {{math|''g''(''μ'')}} as a function of the momentum scale {{mvar|μ}} is determined by the full Gell-Mann–Low equation
{{NumBlk|:|<math>\frac{dg}{d \ln \mu} =\beta(g)=\beta_2 g^2+\beta_3 g^3+\ldots ~,</math>|{{EquationRef|3}}}}
which gives Eqs.({{EquationNote|1}}),({{EquationNote|2}}) if it is integrated under conditions {{math|1=''g''(''μ'') = ''g''<sub>obs</sub>}} for {{math|1=''μ'' = ''m''}} and {{math|1=''g''(''μ'') = ''g''<sub>0</sub>}} for {{math|1=''μ'' = Λ}}, when only the term with <math>\beta_2</math> is retained in the right hand side.

The general behavior of <math>g(\mu)</math> relies on the appearance of the function {{math|''β''(''g'')}}. According to the classification by Bogoliubov and Shirkov,<ref>
{{cite book
 |author1=N. N. Bogoliubov |author2=D. V. Shirkov |year=1980
 |edition=3rd
 |title=Introduction to the Theory of Quantized Fields
 |publisher={{wipe|John Wiley & Sons}}
 |isbn=978-0-471-04223-5
}}</ref> there are three qualitatively different situations:

{{Ordered list|list-style-type=lower-alpha
|if <math>\beta(g)</math> has a zero at the finite value {{math|''g''<sup>*</sup>}}, then growth of {{mvar|g}} is saturated, i.e. <math>g(\mu)\to g^*</math> for <math>\mu\to\infty</math>;
|if <math>\beta(g)</math> is non-alternating and behaves as <math>\beta(g) \propto g^\alpha</math> with <math>\alpha\le 1</math> for large <math>g</math>, then the growth of <math>g(\mu)</math> continues to infinity;
|if <math>\beta(g) \propto g^\alpha</math> with <math>\alpha > 1</math> for large <math>g</math>, then <math>g(\mu) </math> is divergent at finite value <math>\mu_0</math> and the real Landau pole arises: the theory is internally inconsistent due to indeterminacy of <math>g(\mu)</math> for <math>\mu > \mu_0</math>.
}}

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 {{math|''g''<sub>obs</sub>}} is finite, the theory is internally inconsistent. The only way to avoid it, is to tend <math>\mu_0</math> to infinity, which is possible only for {{math|''g''<sub>obs</sub> → 0}}.

==Conclusions==

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.<ref>{{Cite journal 
| last1 = Callaway | first1 = D. 
| last2 = Petronzio | first2 = R. 
| doi = 10.1016/0550-3213(87)90657-2 
| title = Is the standard model Higgs mass predictable? 
| journal = Nuclear Physics B
| volume = 292 
| pages = 497–526 
| year = 1987 
|bibcode = 1987NuPhB.292..497C | url = https://cds.cern.ch/record/172532 
}}</ref><ref>
{{cite journal
 |author=I. M. Suslov
 |year=2010
 |title=Asymptotic Behavior of the ''β'' Function in the ''φ''<sup>4</sup> Theory: A Scheme Without Complex Parameters |journal=Journal of Experimental and Theoretical Physics
 |volume=111
 |issue=3 |pages=450–465
 |doi=10.1134/S1063776110090153|arxiv=1010.4317
|bibcode = 2010JETP..111..450S |s2cid=118545858
 }}</ref><ref>
{{cite conference |url=http://pos.sissa.it/archive/conferences/117/039/FacesQCD_039.pdf |title=Mapping theorem and Green functions in Yang-Mills theory |last1=Frasca |first1=Marco |year=2011 |conference=The many faces of QCD |conference-url=http://sites.google.com/site/facingqcd/ |publisher=Proceedings of Science |pages=039 |location=Trieste |arxiv=1011.3643 |access-date=2011-08-27|bibcode=2010mfq..confE..39F }}</ref>
<ref>{{Cite journal | doi = 10.1016/0550-3213(84)90410-3| title = Non-triviality of gauge theories with elementary scalars and upper bounds on Higgs masses| journal = Nuclear Physics B| volume = 233| issue = 2| pages = 189–203| year = 1984| last1 = Callaway | first1 = D. J. E. |bibcode = 1984NuPhB.233..189C | url = https://cds.cern.ch/record/146517}}</ref><ref>{{Cite journal 
| last1 = Lindner | first1 = M. 
| doi = 10.1007/BF01479540
| title = Implications of triviality for the standard model
| journal = Zeitschrift für Physik C
| volume = 31
| issue = 2 
| pages = 295–300 
| year = 1986
|bibcode = 1986ZPhyC..31..295L| s2cid = 123166350 
}}
</ref><ref>Urs Heller, Markus Klomfass, Herbert Neuberger, and Pavlos Vranas, (1993).
"Numerical analysis of the Higgs mass triviality bound", ''Nucl. Phys.'', '''B405''': 555-573.</ref>

==See also==
* Hierarchy problem

== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|2}}

[[Category:Renormalization group]]
[[Category:Quantum mechanics]]
[[Category:Mathematical physics]]
[[Category:Physical phenomena]]

{{Author|Harold Foppele}}

{{Sourceattribution|Physics:Quantum triviality|1}}