Landau Pole - Brief History

Brief History

According to Landau, Abrikosov, Khalatnikov, the relation of the observable charge with the “bare” charge for renormalizable field theories is given by expression

where is the mass of the particle, and is the momentum cut-off. For finite and the observed charge tends to zero and the theory looks trivial. 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 :

With growth of the bare charge increases and turn to infinity at the point

The latter singularity is the Landau pole. In fact, the growth of invalidates Eqs.1,2 in the region (since they were obtained for ) and reality of the Landau pole becomes doubtful.

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 .

Landau and Pomeranchuk tried to justify the possibility (c) in the case of QED and theory. They have noted that the growth of in Eq.1 drives the observable charge to the constant limit, which does not depend on . The same behavior can be obtained from the functional integrals, omitting the quadratic terms in the action. If neglecting the quadratic terms is valid already for, it is all the more valid for of the order or greater than unity : it gives a reason to consider Eq.1 to be valid for arbitrary . Validity of these considerations on the quantitative level is excluded by non-quadratic form of the -function. Nevertheless, they can be correct qualitatively. Indeed, the result can be obtained from the functional integrals only for, while its validity for, based on Eq.1, may be related with other reasons; for this result is probably violated but coincidence of two constant values in the order of magnitude can be expected from the matching condition. The Monte Carlo results seems to confirm the qualitative validity of the Landau–Pomeranchuk arguments, though a different interpretation is also possible.

The case (c) in the Bogoliubov and Shirkov classification 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 . It is a widespread opinion, that QED and theory are trivial in the continuum limit. In fact, available information confirms only “Wilson triviality”, which is equivalent to positiveness of for and can be considered as firmly established. Indications of “true” quantum triviality are not numerous and allow different interpretation.