Quantum Triviality - Triviality and The Renormalization Group

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” charge

(1)

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 :

(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, there are three qualitatively different situations:

1. if has a zero at the finite value, then growth of is saturated, i.e. for ;
2. if is non-alternating and behaves as with for large, then the growth of continues to infinity;
3. 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 stronger 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 and the implications for the Standard Model and the resulting Higgs Boson mass bounds have been discussed in .