Landau Pole - Recent Developments

Recent Developments

Solution of the Landau pole problem requires calculation of the Gell-Mann–Low function at arbitrary and, in particular, its asymptotic behavior for . This problem is very difficult and was considered as hopeless for many years: by diagrammatic calculations one can obtain only few expansion coefficients, which do not allow to investigate the function in the whole. The progress became possible after development of the Lipatov method for calculation of large orders of perturbation theory : now one can try to interpolate the known coefficients with their large order behavior and to sum the perturbation series. The first attempts of reconstruction of the function witnessed on triviality of theory. Application of more advanced summation methods gave the exponent in the asymptotic behavior a value close to unity. The hypothesis for the asymptotics was recently confirmed analytically for theory and QED . Together with positiveness of, obtained by summation of series, it gives the case (b) of the Bogoliubov and Shirkov classification, and hence the Landau pole is absent in these theories. Possibility of omitting the quadratic terms in the action suggested by Landau and Pomeranchuk is not confirmed.