Quantum triviality

Encyclopedia

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

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.

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. 1 & 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

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, there are three qualitatively 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 seen by a reductio ad absurdum

. Indeed, if is finite, the theory is internally inconsistent. The only way to avoid it, is to 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 direct physical sense, being related to the lattice spacing and the coefficient in the effective Landau Hamiltonian. The trivial theory with is obtained in the limit , 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 ? 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 -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 favor of its positive solution has appeared recently

Quantum field theory

Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

, 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 bosonHiggs boson

The Higgs boson is a hypothetical massive elementary particle that is predicted to exist by the Standard Model of particle physics. Its existence is postulated as a means of resolving inconsistencies in the Standard Model...

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

Standard Model

The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

of particle physics

Particle physics

Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

, the question of triviality in Higgs models is of great importance.

This Higgs triviality is similar to the Landau pole

Landau pole

In physics, the Landau pole is the momentum scale at which the coupling constant of a quantum field theory becomes infinite...

problem in quantum electrodynamics

Quantum electrodynamics

Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

, 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

LHC

LHC may refer to:* Large Hadron Collider, a particle accelerator and collider located on the Franco-Swiss border near Geneva, SwitzerlandLHC also may refer to:* La hora Chanante, a Spanish comedy television show...

. In these Higgs theories, the interactions of the Higgs particle with itself are posited to generate the masses of the W and Z bosons

W and Z bosons

The W and Z bosons are the elementary particles that mediate the weak interaction; their symbols are , and . The W bosons have a positive and negative electric charge of 1 elementary charge respectively and are each other's antiparticle. The Z boson is electrically neutral and its own...

, as well as lepton

Lepton

A lepton is an elementary particle and a fundamental constituent of matter. The best known of all leptons is the electron which governs nearly all of chemistry as it is found in atoms and is directly tied to all chemical properties. Two main classes of leptons exist: charged leptons , and neutral...

masses like those of the electron

Electron

The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

and muon

Muon

The muon |mu]] used to represent it) is an elementary particle similar to the electron, with a unitary negative electric charge and a spin of ½. Together with the electron, the tau, and the three neutrinos, it is classified as a lepton...

. 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 with the “bare” chargewhere 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. 1 & 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

Renormalization group

In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...

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, there are three qualitatively 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 seen by a reductio ad absurdum

Reductio ad absurdum

In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...

. Indeed, if is finite, the theory is internally inconsistent. The only way to avoid it, is to 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 direct physical sense, being related to the lattice spacing and the coefficient in the effective Landau Hamiltonian. The trivial theory with is obtained in the limit , 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 ? 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

Lattice field theory

In physics, lattice field theory is the study of lattice models of quantum field theory, that is, of field theory on a spacetime that has been discretized onto a lattice. Although most lattice field theories are not exactly solvable, they are of tremendous appeal because they can be studied by...

confirmed Wilson’s conclusion.

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 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

Standard Model

The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

of particle physics

Particle physics

Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

is nontrivial (and whether elementary scalar Higgs particles can exist) remains an important unresolved question. The evidence in favor of its positive solution has appeared recently