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Summation of the perturbation series for the Gell-Mann--Low function \beta(g)of \phi^4 theory leads to the asymptotics \beta(g)=\beta_\infty g^\alpha atg\to\infty, where \alpha\approx 1 for space dimensions d=2,3,4. The naturalhypothesis arises, that asymptotic behavior is \beta(g) \sim g for all d.Consideration of the "toy" zero-dimensional model confirms the hypothesis andreveals the origin of this result: it is related with a zero of a certainfunctional integral. This mechanism remains valid for arbitrary spacedimensionality d. The same result for the asymptotics is obtained forexplicitly accepted lattice regularization, while the use of high-temperatureexpansions allows to calculate the whole \beta-function. As a result, the\beta-function of four-dimensional \phi^4 theory is appeared to benon-alternating and has a linear asymptotics at infinity. The analogoussituation is valid for QED. According to the Bogoliubov and Shirkovclassification, it means possibility to construct the continuous theory withfinite interaction at large distances. This conclusion is in visiblecontradiction with the lattice results indicating triviality of \phi^4 theory.This contradiction is resolved by a special character of renormalizability in\phi^4 theory: to obtain the continuous renormalized theory, there is no needto eliminate a lattice from the bare theory. In fact, such kind ofrenormalizability is not accidental and can be understood in the framework ofWilson's many-parameter renormalization group. Application of these ideas toQCD shows that Wilson's theory of confinement is not purely illustrative, buthas a direct relation to a real situation. As a result, the problem ofanalytical proof of confinement and a mass gap can be considered as solved, atleast on the physical level of rigor.

A thorny path of field theory: from triviality to interaction and confinement