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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 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 boson and Z boson, 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

Mathematically speaking, the existence of a nontrivial field theory typically requires that there be an "ultraviolet" or UV fixed point in the momentum-space renormalization group equations of the theory, corresponding to a zero of the so-called Beta-function. The observable "effective" or "running" coupling at a given momentum scale Q is obtained by solving the renormalization group equation that defines this Beta-function. This equation is sometimes called the Callan–Symanzik equation:

\frac{\partial}{\partial t} g = \beta(g)

where

 t = \ln \left(\frac{Q^2}{\mu^2}\right)

and μ is the momentum scale at which the (measureable) renormalized couplings gR are defined, thus g(t = 0) = gR. The renormalization group equation can easily be generalized to a theory with a number of couplings, such as the coupling of the scalar field to itself, the electric and weak charges, a Yukawa coupling set that generates particle masses, and so on.

The argument for triviality is typically phrased as a reductio ad absurdum in this language. One chooses a given set of renormalized coupling constants gR as initial conditions (t = 0) and integrates the momentum-space renormalization group equations forward in t to evaluate the running couplings at larger values of Q. If unphysical (e.g., infinite) values of the couplings appear at a finite value of Q, then the theory is inconsistent for the chosen set of renormalized couplings, and one must choose another set of initial conditions (renormalized couplings). The process of solving the renormalization group equations thus serves to generate constraints upon the allowed values of the renormalized coupling constants of the theory.

As a practical matter, the use of the momentum-space renormalization group is confined to regions where the coupling constants are small, for then the beta functions can be calculated in perturbation theory. More realistic calculations typically utilize the real-space renormalization group [3] due to Kenneth G. Wilson, as applied (via numerical simulation) to models formulated using lattice gauge theory, also invented by Wilson. The situation regarding 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.

References

  1. ^ R. Fernandez, J. Froehlich, A. D. Sokal, "Random Walks, Critical Phenomena, and Triviality in Quantum Field Theory". Springer (April 1992) ISBN 0387543589
  2. ^ D.J.E. Callaway (1988). "Triviality Pursuit: Can Elementary Scalar Particles Exist?". Phys. Reports 167: 241–320. doi:10.1016/0370-1573(88)90008-7.  
  3. ^ K. G. Wilson, Rev. Mod. Phys. 47, 4(1975), "The renormalization group: critical phenomena and the Kondo problem"
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